MATH 215 C10 - Study Guide: Unit 4

In Unit 3, we discussed probability distributions for both discrete and continuous random variables. At the start of Unit 4, we

examine sampling distributions that refer to probability distributions of sample statistics, such as the sample mean and sample

proportion. Once you understand the concept of sampling distributions, you will be ready to begin the field of inferential

statistics.

The first topic we consider is the Central Limit theorem, which allows us to use the properties of sampling distributions to

construct confidence interval estimates and conduct tests of hypotheses involving population means and proportions.

Confidence interval estimation allows us to estimate a population mean or population proportion based on sample data. As an

example, the owners of a restaurant could estimate the mean age of all of their customers, based on a sample survey. As a

different example, a medical researcher could estimate the proportion of patients who exhibit a specific side-effect when taking a

new drug.

Hypothesis testing is used to test a specific claim about a population based on sample data. For example, a sociologist might

want to test the claim that, on average, those with master’s degrees make more money than those with bachelor’s degrees. A

consumer might want to question a recent advertisement put out by a weight-loss centre that claims to reduce the weight of its

clients by at least 10 pounds within a month. A political strategist might want to challenge the view that the political party

currently in power will win a majority of the votes in the next election.

Unit 4 of MATH 215 consists of the following sections:

4-1 Mean and Standard Deviation of the Sampling Distribution of the Sample Mean

4-2 Shape of the Sampling Distribution of the Sample Mean

4-3 Mean, Standard Deviation, and Shape of the Sampling Distribution of the Sample Proportion

4-4 Estimation of a Population Mean: Population Standard Deviation Is Known

4-5 Estimation of a Population Mean: Population Standard Deviation Is Unknown

4-6 Estimation of a Population Proportion: Large Samples

4-7 Hypothesis Tests about a Single Population Mean: Population Standard Deviation Is Known

4-8 Hypothesis Tests about a Single Population Mean: Population Standard Deviation Is Unknown

4-9 Hypothesis Tests about a Single Population Proportion: Large Samples

The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete

Assignment 4.

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the following key terms:

population distribution

sampling distribution

sampling error and non-sampling error

MATH 215 C10 - Study Guide: Unit 4

1

mean and standard deviation of sampling distributions of the sample mean

2. find the mean and standard deviation of the sampling distribution of the sample mean, given the mean and standarddeviation of the population distribution, and given the sample size.

Read the following sections in Chapter 7 of the textbook:

Chapter 7 Introduction

Section 7.1

Section 7.2

Be prepared to read the material in Chapter 7 at least twice—the first time for a general overview of topics, and the second time

to concentrate on the terms and examples presented. Return to these sections when you need to review these topics.

These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned

textbook readings.

Sample vs sampling distributions (https://www.youtube.com/watch?v=dF18I7-Re4I) (Mine Cetinkaya-Rundel)

Sampling Distributions: Introduction to the Concept (https://www.youtube.com/watch?v=Zbw-YvELsaM&nohtml5=False) (jbstatistics)

Sampling Distributions (https://www.youtube.com/watch?v=DmZJ1blQOns&nohtml5=False) (Brian Foltz)

Variation and Sampling Error (https://www.youtube.com/watch?v=y3A0lUkpAko) (Dr Nic’s Maths and Stats)

Dancing Statistics: explaining the statistical concept of ‘sampling’ and ‘standard error’ through dance(https://www.youtube.com/watch?v=5fGu8hvdZ6s&index=1&list=PLCkLQOAPOtT2H1hJRUxUYOxThRwfVI9jI)(BPSOfficial)

Sampling Distribution of the Sample Mean (https://www.youtube.com/watch?v=q50GpTdFYyI&nohtml5=False)(jbstatistics)

Standard Error of the Mean (https://www.youtube.com/watch?v=uIHFbMn8SBc) (Brandon Foltz)

Note: This is relevant to the situation when a parameter is to be estimated about a population mean from asample. The standard error is the standard deviation of the sampling distribution of a statistic (for example, asample mean or a sample proportion). The term may also be used to refer to an estimate of that standarddeviation, derived from a particular sample used to compute the estimate.

Difference Between Standard Deviation and Standard Error (https://www.youtube.com/watch?v=u4W7_Q1OiYQ)(Tommea Analytics)

Standard Deviation and Standard Error of the Mean (https://www.youtube.com/watch?v=3UPYpOLeRJg) (PiersSupport)

Complete the following exercises from Chapter 7 of the textbook:

Exercises 7.11, 7.15, and 7.17 on page 283

Remember to show your work as you develop your answers.

Solutions to these exercises are provided in the Student Solutions Manual for Chapter 7 (interactive textbook) and on


2

pages AN10 and AN11 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Remember, it is very important that you make a concerted effort to answer each question independently before you refer to the

solutions. If your answers differ from those provided and you cannot understand why, contact your tutor for assistance.


1. state the Central Limit theorem and apply it to problems involving sample means.

2. determine the shape of the sampling distribution of the sample mean, given information about the populationdistribution, the sample size, or both.

3. find the probability that the value of the sample mean will fall within a specified interval, given the population mean, thepopulation standard deviation and the sample size.


Section 7.3

Section 7.4


textbook readings.

Properties of a Sampling Distribution (https://www.youtube.com/watch?v=tswINqdruQ4) (Steve Mays)

An Introduction to the Central Limit Theorem (https://www.youtube.com/watch?v=Pujol1yC1_A) (jbstatistics)

AP Statistics: Sampling Distributions & the Central Limit Theorem (https://www.youtube.com/watch?v=0OSS-IWcfzQ)(Math with Mark)

Sampling Distributions & the Central Limit Theorem (https://www.youtube.com/watch?v=iM0MMgH5cLw&nohtml5=False) (Jennifer Edmonds)

The videos suggested above for Section 7.2 also relate to Section 7.4.

Complete the following exercises from Chapter 7 of the textbook (page numbers are for the downloadable eText):

Exercises 7.23 and 7.25 on page 288

Exercise 7.27 on page 289

Exercises 7.33, 7.35, 7.37, 7.39, and 7.41 on page 293

Solutions are provided in the Student Solutions Manual for Chapter 7 (interactive textbook) and on page AN11 in the Answers to

Selected Odd-Numbered Exercises section (downloadable eText).


3



population proportion and sample proportion

sampling distribution of the sample proportion

mean and standard deviation of the sampling distribution of the sample proportion

Central Limit theorem for sample proportions

2. determine the mean, standard deviation and shape of the sampling distribution of the sample proportion, given thepopulation proportion and the sample size.

3. find the probability that the value of the sample proportion will fall within a specified interval, given the populationproportion and the sample size.


Section 7.5

Section 7.6


textbook readings.

The Sampling Distribution of the Sample Proportion (https://www.youtube.com/watch?v=fuGwbG9_W1c&list=UUiHi6xXLzi9FMr9B0zgoHqA&index=6) (jbstatistics)

AP Statistics: Sampling Distributions for Proportions (https://www.youtube.com/watch?v=oDw6mbeDy-M) (MichaelPorinchak)

1. Complete the following exercises from Chapter 7 of the textbook (pages numbers are for the downloadable eText):


Exercises 7.63 and 7.65 on pages 301–302

2. Complete the Self-Review Test for Chapter 7 (pages 304–305 of the downloadable eText).

Solutions are provided in the Student Solutions Manual for Chapter 7 (interactive textbook) and on page AN11 in theAnswers to Selected Odd-Numbered Exercises section (downloadable eText).


4

For extra practice with the material presented in this section, you can complete the following questions and exercises, for which

the solutions are provided in the textbook:

1. Any odd-numbered chapter-section practice questions that are not assigned above

2. The odd-numbered Supplementary Exercises and Advanced Exercises at the end of Chapter 7 (pages 303–304 of thedownloadable eText)



point estimates and interval estimates

significance level

confidence level and confidence interval

margin of error

2. use the distribution to construct a confidence interval for the population mean when the population standard deviationis known, the population distribution is normal and the sample size is small ( ).

3. use the distribution to construct a confidence interval for the population mean when the population standard deviationis known and the sample size is large ( ).

4. compute the sample size that will be required to estimate the mean, given the confidence level, the population standarddeviation and a specified margin of error.



Section 8.1

Section 8.2

Be prepared to read the material in Chapter 8 at least twice—the first time for a general overview of topics, and the second time

to concentrate on the terms and examples presented. Return to these sections when you need to review these topics.


textbook readings.

Sample Mean Proximity to Population Mean (https://www.youtube.com/watch?v=QbGRTWLjp7c&nohtml5=False)(Brandon Foltz)


5

Understanding Confidence Intervals (https://www.youtube.com/watch?v=tFWsuO9f74o) (Dr Nic’s Maths and Stats)

Point Estimators (https://www.youtube.com/watch?v=4v41z3HwLaM) (Brandon Foltz)

Introduction to Confidence Intervals (https://www.youtube.com/watch?v=wdsDz_2cEzw&nohtml5=False) (KennethStrazzeri)

Margin of Error and Confidence Intervals (https://www.youtube.com/watch?v=9-wvUL-keyc&nohtml5=False) (KennethStrazzeri)

Confidence Interval Assumptions (https://www.youtube.com/watch?v=Z9XisOHaZ-c&nohtml5=False) (KennethStrazzeri)

Scores – Statistics (https://www.youtube.com/watch?v=NY2zWGBXBhU) (Math Meeting) [for when is known]

Confidence Intervals for a Population Mean ( known) (https://www.youtube.com/watch?v=czdwHU27OqA) (JoshuaEmmanuel)

Confidence Interval Estimation, Sigma Known (https://www.youtube.com/watch?v=9GtaIHFuEZU) (Brandon Foltz)

Intro to Confidence Intervals for One Mean (Sigma Known) (https://www.youtube.com/watch?v=KG921rfbTDw)(jbstatistics)


Exercises 8.11, 8.13, 8.15, 8.19, 8.23, and 8.25 on pages 322–323




distribution

sample standard deviation

2. use the distribution to construct a confidence interval for the population mean when the population standard deviationis unknown, the population distribution is normal and the sample size is small ( ).

3. use the distribution to construct a confidence interval for the population mean when the population standard deviationis unknown and the sample size is large ( ).

Read Section 8.3 in Chapter 8 of the downloadable eText.



6

textbook reading.

Scores – Statistics (https://www.youtube.com/watch?v=_x6j-rH33Og) (Math Meeting) [for when is unknown]

Confidence Interval Assumptions (https://www.youtube.com/watch?v=Z9XisOHaZ-c&nohtml5=False) (KennethStrazzeri)

Confidence Intervals from Repeated Samples (sigma unknown) (https://www.youtube.com/watch?v=MEqUHfEnRJ4)(Kenneth Strazzeri)

Confidence Intervals for One Mean: Sigma Not Known ( Method) (https://www.youtube.com/watch?v=bFefxSE5bmo)(jbstatistics)

Confidence Interval Concepts, Sigma Unknown (https://www.youtube.com/watch?v=BQ88ni4bJNA) (Brandon Foltz)

Confidence Interval Problems, Sigma Unknown (https://www.youtube.com/watch?v=jHfMQB4HCkQ) (Brandon Foltz)




Solutions are provided in the Student Solutions Manual for Chapter 8 (interactive textbook) and on page AN12 in the Answers to

Selected Odd-Numbered Exercises section (downloadable eText).


1. define and apply the “estimator of the standard deviation of the sampling distribution of the sample proportion.”

2. use the distribution to construct a confidence interval for the population proportion, given sample data.

3. compute the sample size that will be required to estimate the proportion, given the level of confidence and a specifiedmargin of error.

Read Section 8.4 in Chapter 8 of the textbook.


textbook reading.

Confidence Intervals for Categorical Data (https://www.youtube.com/watch?v=qzw-AA_Erxc&nohtml5=False) (KennethStrazzeri)

Confidence Intervals for a Proportion: Determining the Minimum Sample Size (https://www.youtube.com/watch?v=mmgZI2G6ibI&index=5&list=UUiHi6xXLzi9FMr9B0zgoHqA) (jbstatistics)

1. Complete the following exercises from Chapter 8 of the textbook (page numbers are for the downloadable eText):


7

Exercise 8.53 on page 335

Exercises 8.57, 8.59, 8.61, and 8.63 on page 336


Supplementary Exercises 8.77, 8.79, 8.81, 8.83, and 8.85 on pages 338–339

2. Complete the Self-Review Test for Chapter 8 (pages 339–340 of the downloadable eText). Omit questions 14 and 15.




1. Any odd-numbered chapter-section practice questions and Supplementary Exercises that are not assigned above

2. The odd-numbered Advanced Exercises at the end of Chapter 8 (page 339 in the eText)



null hypothesis

alternative hypothesis

critical value

Type I error

level of significance

Type II error

two-tailed test

left-tailed test

right-tailed test

test statistic or observed value

statistically significantly different and statistically not significantly different

2. use the critical value approach to perform a hypothesis test about the population mean, given the population standard


8

deviation and sample data.

3. use the approach to perform a hypothesis test about the population mean, given the population standarddeviation and sample data.

1. Read the following sections in Chapter 9 of the textbook:


Sections 9.1

Section 9.2

2. Read Additional Topics 4A, 4B, and 4C in this Study Guide, below.

Important: Complete this reading before you complete the exercises for this section.

Be prepared to read the material in Chapter 9 and the additional topics at least twice—the first time for a general overview of

topics, and the second time to concentrate on the terms and examples presented. Return to these sections when you need to

review these topics.


textbook readings.

Intro to Hypothesis Testing - an Overview (https://www.youtube.com/watch?v=o1IRoYSZIJc) (Kevin Martz)

Choosing Which Statistical Test to Use (https://www.youtube.com/watch?v=rulIUAN0U3w) (Dr Nic’s Maths and Stats)

Relationship Between Hypothesis Tests and Confidence Intervals (https://www.youtube.com/watch?v=k1at8VukIbw&list=PLvxOuBpazmsNo893xlpXNfMzVpRBjDH67&index=9) (two-sided) (jbstatistics)

Intro to Hypothesis Testing in Statistics – Problems & Examples (https://www.youtube.com/watch?v=VK-rnA3-41c)(Math and Science)

An Introduction to Hypothesis Testing (https://www.youtube.com/watch?v=tTeMYuS87oU&index=1&list=PLvxOuBpazmsNo893xlpXNfMzVpRBjDH67) (jbstatistics)

One-Tailed and Two-Tailed Hypotheses (https://www.youtube.com/watch?v=MOg4XeW2oLA) (Roger Morrissette)

One-Sided Test or Two-Sided Test? (https://www.youtube.com/watch?v=VP1bhopNP74) (jbstatistics)

Understanding the (https://www.youtube.com/watch?v=eyknGvncKLw&nohtml5=False) (Dr Nic’s Maths andStats)

What is a ? (https://www.youtube.com/watch?v=HTZ8YKgD0MI) (jbstatistics)

Important Statistical Concepts: significance, strength, association, causation (https://www.youtube.com/watch?v=FG7xnWmZlPE) (Dr Nic’s Maths and Stats)

Visualizing Type I and Type II Errors (https://www.youtube.com/watch?v=k80pME7mWRM) (Brandon Foltz)

Type I and Type II Errors (https://www.youtube.com/watch?v=VFMcGdWp0MQ) (Brandon Foltz)

Type I Errors, Type II Errors and the Power of the Test (https://www.youtube.com/watch?v=7mE-K_w1v90)(jbstatistics)

What Factors Affect the Power of a test? (https://www.youtube.com/watch?v=K6tado8Xcug&list=PLvxOuBpazmsNo893xlpXNfMzVpRBjDH67&index=12) (jbstatistics)

Calculating the Power and the Probability of a Type II Error (one-tailed -test example) (https://www.youtube.com/watch?v=BJZpx7Mdde4) (jbstatistics)

Calculating the Power and the Probability of a Type II Error (two-tailed -test example) (https://www.youtube.com


9

/watch?v=NbeHZp23ubs) (jbstatistics)

To or to , That is the Question (https://www.youtube.com/watch?v=Kzqm8F9Le_4) (Brandon Foltz)

versus (https://www.youtube.com/watch?v=YsalXF5POtY&nohtml5=False) (Math Meeting)

Hypothesis Tests on One Mean: or ? (https://www.youtube.com/watch?v=vw2IPZ2aD-c) (jbstatistics)

for One Mean: Introduction (https://www.youtube.com/watch?v=pGv13jvnjKc) (jbstatistics) [for when isknown]

Single Sample Hypothesis Concepts (https://www.youtube.com/watch?v=HoqzIR8xj4s) (Brandon Foltz)

Single Sample Hypothesis Examples (https://www.youtube.com/watch?v=tsPv-ffN-0M) (Brandon Foltz)

Single Sample Hypothesis Alpha and (https://www.youtube.com/watch?v=i8xChxbZ3uQ) (BrandonFoltz)

This video gives a relative comparison of differing alpha values and their corresponding critical and of the test statistic.

Calculate the in Statistics – Formula to Find the in Hypothesis Testing (https://www.youtube.com/watch?v=KLnGOL_AUgA) (Math and Science)

One helpful comment from this video is “If your is low, reject the .”

What is a ? (https://www.youtube.com/watch?v=UsU-O2Z1rAs) (jbstatistics)

for One Mean: the (https://www.youtube.com/watch?v=m6sGjWz2CPg) (jbstatistics)

for One Mean: an example (https://www.youtube.com/watch?v=Xi33dGcZCA0&index=5&list=PLvxOuBpazmsNo893xlpXNfMzVpRBjDH67) (jbstatistics)

Hypothesis Testing ( Method) (https://www.youtube.com/watch?v=cW16A7hXbTo) (poysermath)

Hypothesis Testing using the Method (Example) (https://www.youtube.com/watch?v=iItTzRRmsIo)(poysermath)

A Tool for Calculating the for Various Hypothesis Tests (http://www.statdistributions.com/normal/)(statdistributions.com)

Statistical Significance versus Practical Significance (https://www.youtube.com/watch?v=_k1MQTUCXmU&index=8&list=PLvxOuBpazmsNo893xlpXNfMzVpRBjDH67) (jbstatistics)

for One Mean: The Rejection Region Approach (https://www.youtube.com/watch?v=60x86lYtWI4) (jbstatistics)

Note: The rejection region approach is another name for the critical value approach

Determining Critical Values and Rejection Regions (https://www.youtube.com/watch?v=s9Jwe4AcraA) (Clutch Prep)

Stats: Hypothesis Testing (Traditional Method) (https://www.youtube.com/watch?v=rWFDXt-MlNs) (poysermath)

Once you have completed all the reading, including Additional Topics 4A, 4B, and 4C, complete the following exercises from

Chapter 9 of the textbook (page numbers are for the downloadable eText):



Exercises 9.25, 9.27, 9.29, and 9.31 on page 366

Solutions are provided in the Student Solutions Manual for Chapter 9 (interactive textbook) and on pages AN12 and AN13 in

the Answers to Selected Odd-Numbered Exercises section (downloadable eText).


10

Unless otherwise stated in the exercise/problem you are working on, make sure that you show your work regarding all four

steps in the approach, as follows:

Step 1: State the null hypothesis ( ) and the alternative hypothesis ( ).

Step 2: Select the distribution to use.

Step 3: Calculate the .

Step 4: Make a decision.

The (or probability value) is the probability of getting a sample statistic (such as the sample mean or its related value)

or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true.

For a one-tailed test, the is given by the area in the tail of the sampling distribution curve beyond the observed value of

the sample statistic (or its related value).

The figure below, reproduced from your text, shows the for a right-tailed test about , where has a “ ” sign.

Mann explains that “for a left-tailed test, the will be the area in the lower tail of the sampling distribution curve to the

left of the observed value” of the sample mean (or ) (Mann 356).

Step 4 comprises two components. You must complete both of these components in order to complete Step 4 and state your

decision properly.

Note: you will not receive any marks for completing Step 4 of a hypothesis test unless you complete both of

these components.

For the given problem/exercise, display a comparison of the computed from Step 3 with the given level of significance.

Based on this comparison, state “reject the null hypothesis” or “do not reject the null hypothesis” by applying the following rule:

If the , reject .

If the , do not reject .

To further explain the rule above, if the is relatively low, this means that the probability of generating the sample mean

observed in the problem is low, assuming that is true. More likely, is not true, so the null hypothesis should be rejected.


11

Based on your decision to reject or not reject the null hypothesis, state the conclusion in terms of the practical context of the

problem/exercise at hand. For example, if the test of hypothesis relates to average income, your stated conclusion should be in

terms of average income; or, if the test of hypothesis relates to mean weight, your stated conclusion should be in terms of mean

weight.

Unless otherwise stated in the exercise/problem you are working on, make sure that you show your work regarding all five

steps in the critical-value approach, as follows:



Step 3: Determine the rejection and non-rejection regions (critical values, etc.).

Step 4: Calculate the value of the test statistic.


We strongly encourage you to sketch the appropriate graph illustrating the rejection and non-rejection regions, as this will help

you to correctly determine the critical values.

Step 5 comprises two components. You must complete both of these components in order to complete Step 5 and state your

decision properly.

Note: you will not receive any marks for completing Step 5 of a hypothesis test unless you complete both of

these components.

For the given problem/exercise, display a comparison of the computed test statistic from Step 4 with the determined

rejection/non-rejection regions in Step 3. Based on this comparison, state “reject the null hypothesis” or “do not reject the null

hypothesis” by applying the following rule:

If the test statistic falls inside the rejection region, reject .

If the test statistic falls outside the rejection region, do not reject .

Based on your decision to reject or not reject the null hypothesis, state the conclusion in terms of the practical context of the

problem/exercise at hand. For example, if the test of hypothesis relates to average income, your stated conclusion should be in

terms of average income; or, if the test of hypothesis relates to mean weight, your stated conclusion should be in terms of mean

weight.

There is a one-to-one correspondence between the approach to hypothesis testing and the critical value approach:

if the for a test of hypothesis is less than α , then the observed value of the test statistic will fall in the rejectionregion of the critical value approach, and consequently will be rejected;

if the observed value of the test statistic falls in the rejection region of the critical value approach, then the will beless than α , and again will be rejected.


12

Most statistical software packages perform tests of hypotheses using a approach rather than a critical value approach.

Our experience has shown, however, that students find the critical value approach more “user-friendly” (i.e., understandable)

than the approach.

An advantage of using the approach rather than the critical value approach is that with this approach you are able not

only to decide whether to reject or not reject , but also to get a sense of how significant the decision/conclusion is (that is, how

strong the evidence is to support the decision to reject or not reject ). This is further explained below.

The following table provides guidelines to interpreting when you encounter them in future research.

In essence, a null hypothesis ( ) is a claim that is “on trial.” It represents the status quo in a given situation, which is

considered innocent until proven guilty beyond a reasonable doubt. In medical research, an may be that a drug or treatment

has “no” effect; in business research, an may be that an advertising program has “no” effect. As the table above shows, very

small provide strong evidence that a drug or treatment does have an effect, or that an advertising program is indeed

effective after all.

After completing the readings and exercises for this section, you should be able to use the critical value approach to perform a

hypothesis test about the population mean, given sample data, when the population standard deviation is unknown.

1. Read Section 9.3 in Chapter 9 of the textbook.

2. Read Additional Topics 4D and 4E in this Study Guide, below.

Important: Complete this reading before you complete the exercises for this section.


textbook reading.


13

Introduction to the Distribution (non-technical) (http://www.youtube.com/watch?v=Uv6nGIgZMVw&feature=youtu.be) (jbstatistics)

An Introduction to the Distribution (mathematical) (http://www.youtube.com/watch?v=T0xRanwAIiI) (jbstatitics)

for One Mean: Introduction (https://www.youtube.com/watch?v=T9nI6vhTU1Y&index=14&list=PLvxOuBpazmsNo893xlpXNfMzVpRBjDH67) (jbstatistics)

for One Mean: investigating the normality assumption (https://www.youtube.com/watch?v=U1O4ZFKKD1k&list=PLvxOuBpazmsNo893xlpXNfMzVpRBjDH67&index=16) (jbstatistics)

Single Sample Hypothesis Concepts (https://www.youtube.com/watch?v=NQWZefn41VY) (Brandon Foltz)

Single Sample Hypothesis Examples (https://www.youtube.com/watch?v=dDsKP7wVpzM&list=PLIeGtxpvyG-IZRHcZcOy12jp7ywuRbE7l&index=11) (Brandon Foltz)

for One Mean: an example (https://www.youtube.com/watch?v=kQ4xcx6N0o4&list=PLvxOuBpazmsNo893xlpXNfMzVpRBjDH67&index=15) (jbstatistics)

Note: The content of this video is relevant only up to the 7:35 minute mark.


Exercises 9.35, 9.37, 9.41, 9.43, 9.45, and 9.49 on pages 374–375


This reading takes Example 9-5 from the textbook, which you have already read, and adds a more detailed explanation of

estimating the for a two-tailed test, as opposed to a similar test involving the distribution.

EXAMPLE 9-5: Age at Which Children Start Walking

A psychologist claims that the mean age at which children start walking is 12.5 months. Carol wanted to check

if this claim is true. She took a random sample of 18 children and found that the mean age at which these

children started walking was 12.9 months with a standard deviation of .80 month. It is known that the ages at

which all children start walking are approximately normally distributed. Find the for the test that the

mean age at which all children start walking is different from 12.5 months. What will your conclusion be if the

significance level is 1%?

Solution: Let be the mean age at which all children start walking, and let be the corresponding mean for

the sample. From the given information,

, , and

The claim of the psychologist is that the mean age at which children start walking is 12.5 months.

[Source: Prem S. Mann, Introductory Statistics, 9th ed. (Wiley, 2016) [VitalSource], 368–369. This material

is reproduced with the permission of John Wiley & Sons Canada, Ltd.]

To test the hypothesis and to make the decision, we apply the following four steps:

Step 1. State the null and alternative hypotheses.

: (The mean walking age is 12.5 months.)


14

: (The mean walking age is different from 12.5 months.)

Step 2. Select the distribution to use.

In this example, we do not know the population standard deviation , the sample size is small ( ),

and the population is approximately normally distributed. Therefore, we will use the distribution to find

the for this test.

Step 3. Calculate the .

The sign in the alternative hypothesis indicates that the test is two-tailed. To find the , first we

find the degrees of freedom and the value for . Then, the is equal to twice the

area in the tail of the distribution curve beyond this value for



The value (also called the test statistic) is:

, so (two-tailed)

The is the area under the distribution curve beyond “ , ” which is , as shown below:

In determining the related to , the best thing to do is to use Table V in Appendix B of the textbook to find the range that

contains the (i.e., estimate the ), as explained below.


15

Table V The Distribution Table

The entries in this table give the critical values

of for the specified number of degrees

of freedom and areas in the right tail.

a. Read down the Distribution Table (above) until you find the appropriate degrees of freedom, which in this case are:.

b. Locate the calculated value of 2.121 in the row with 17 degrees of freedom. It falls between 2.110 and 2.567.

c. Read to the top of the table to locate the area to the right of this calculated value. The area to the right is between 0.025and 0.01. This range of area is one-half the desired , because this is a two-tailed hypothesis test.

d. Since we have a two-tailed test, multiply this range of areas by two to get the range of the desired , as follows:

Estimated : is between 2(0.01) and 2(0.025)Estimated : is between 0.02 and 0.05

Step 4. Make a decision.

Since the estimated exceeds , we do not reject . Therefore, we cannot conclude that the mean walking

age is different from 12.5 months.


16

This reading takes Example 9-6 from the textbook, which you have already read, and adds a more detailed explanation of

estimating the for a one-tailed test, as opposed to a similar test involving the distribution.

EXAMPLE 9-6: Life of Batteries

Grand Auto Corporation produces auto batteries. The company claims that its top-of-the-line Never Die

batteries are good, on average, for at least 65 months. A consumer protection agency tested 45 such batteries

to check this claim. It found that the mean life of these 45 batteries is 63.4 months, and the standard

deviation is 3 months. Find the for the test that the mean life of all such batteries is less than

65 months. What will your conclusion be if the significance level is 2.5%?

Solution: Let be the mean life of all such auto batteries, and let be the corresponding mean for the

sample. From the given information,

, , and

The claim of the company is that the mean life of these batteries is at least 65 months. [To conduct the test of

hypothesis] and make the decision, we apply the following four steps.

Step 1. State the null and alternative hypotheses.

We are to test if the mean life of these batteries is at least 65 months. Hence, the null and alternative

hypotheses are

: (The mean life of batteries is at least 65 months.)

: (The mean life of batteries is less than 65 months.)

Step 2. Select the distribution to use.

In this example, we do not know the population standard deviation , and the sample size is large (

). [. . .] Consequently, we will use the distribution to find the for this test.

Step 3. Calculate the .

The sign in the alternative hypothesis indicates that the test is left-tailed. To find the , first we

find the degrees of freedom and the value for . Then, the is given by the area in

the tail of the distribution curve beyond this value for .



The value (that is, the test statistic, or simply ) is:

The is the area under the distribution curve beyond , as shown below:

From


17

In determining the related to , the best thing to do is to use Table V in Appendix B in the textbook to find the range that

contains the (i.e., estimate the ) as explained below.

Table V The Distribution Table (continued)

The entries in this table give the critical values

of for the specified number of degrees

of freedom and areas in the right tail.

a. Read down the Distribution Table (above) until you find the appropriate degrees offreedom, which in this case are:.

b. Ignoring the sign of the calculated test statistic, locate it in the row with 44 degrees of freedom. It falls to the right of3.286.

c. Read to the top of the table to locate the area to the right of this calculated value. The area to the right is less than 0.001.Since the distribution is symmetric, the area to the left of is also less than 0.001.

d. Since we have a one-tailed test, this estimated area to the left of is our estimated . That is, .

Step 4. Make a decision.


18

Since the estimated of 0.001 is less than , we reject . Therefore, we can conclude that the mean life of

such batteries is less than 65 months.

After completing the readings and exercises for this section, you should be able to do the following: Use the critical value

approach and the approach to perform a hypothesis test about the population proportion, given data from a large

sample.

Read Section 9.4 in Chapter 9 of the textbook.


textbook reading.

An Introduction to Inference for a Proportion (https://www.youtube.com/watch?v=owYtDtmrCoE&list=UUiHi6xXLzi9FMr9B0zgoHqA&index=8) (jbstatistics)

Inference for One Proportion: an example for a Confidence Interval and a Hypothesis Test (https://www.youtube.com/watch?v=M7fUzmSbXWI&index=7&list=UUiHi6xXLzi9FMr9B0zgoHqA) (jbstatistics)

Inference for a Population Proportion (https://www.youtube.com/watch?v=75p4rGBngpo) (Bryan Nelson)

1. Complete the following exercises from Chapter 9 of the textbook (page numbers are for the downloadable eText):

Exercises 9.53, 9.55, 9.57, 9.61, 9.63, and 9.65 on page 383

Supplementary Exercises 9.73, 9.75, 9.79, 9.81, and 9.83 on pages 386–387

Note: For Exercises 9.75, 9.79, and 9.83, use the critical value approach.

2. Complete the problems in the Self-Review Test for Chapter 9 (pages 388–389 of the downloadable eText). If a problemasks you to conduct a test of hypothesis and does not specify which approach to use, use the critical value approach.

Solutions are provided in the Student Solutions Manual for Chapter 9 (interactive textbook) and on pages AN13 andAN14 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

3. Complete the Unit 4 Self-Test below.


19



1. Any odd-numbered chapter-section practice questions and Supplementary Exercises that are not assigned above

2. The odd-numbered Advanced Exercises at the end of Chapter 9 (page 387 in the eText)

Once you have completed the Unit 4 Self-Test below, complete Assignment 4. You can access the assignment in the Assessment

section of the course home page. Once you have completed the assignment, submit it to your tutor for marking using the drop

box on the page for Assignment 4.

The self-test questions are shown here for your information. Download the Unit 4 Self-Test (https://fst-course.athabascau.ca

/science/math/215/r10/self_test/self_test04.html) document and write out your answers. Show all your work and keep your

calculations to four decimal places, unless otherwise stated. You can access the solutions to this self-test on the course home

page.

1. Circle True (T) or False (F) for each of the following:

a. T F The standard deviation of the sampling distribution of the sample mean is equal to the populationstandard deviation.

b. T F If the population distribution is positively skewed, then the sampling distribution of the sample meanis also positively skewed.

c. T F When the population standard deviation is unknown and the sample size exceeds 30, the distribution is used to compute a confidence interval for the population mean.

d. T F When the population standard deviation is known and the sample size exceeds 30, the distribution isused to compute a confidence interval for the population mean.

e. T F A larger sample size will tend to reduce the width of a confidence interval.

f. T F In conducting a test of hypothesis, if the exceeds the level of significance, we reject the nullhypothesis.

g. T F In conducting a test of hypothesis, if the alternative hypothesis consists of a “ ” expression, thecritical value will be a negative number.

h. T F In conducting a test of hypothesis, if the is less than 0.001, the evidence against the nullhypothesis is considered to be very weak.

2. Past census surveys in a large Canadian province indicate that 40% of provincial voters favor the implementation of acarbon tax to combat global warming.

Consider a sampling (random) experiment where 100 voters are selected at random and the sample proportion of voterswho favor a carbon tax is to be observed.

a. What would be the shape of the sampling distribution of the sample proportion in favor of the carbon tax, andwhy?

b. Determine the mean of the sampling distribution of the sample proportion in favor of the carbon tax.

c. Determine the standard deviation of the sampling distribution of the sample proportion in favor of the carbon tax.

d. Find the probability (to 4 decimal places) that, in the random sample of 100 voters, the sample proportion whofavor a carbon tax is:

i. less than 0.30

ii. between 0.30 and 0.40


20

e. If, among the 100 voters selected at random, 44 are in favor of a carbon tax, compute the sampling error. Assumethat there are no non-sampling errors.

3. Recent studies involving all students in a community college found that these students spend an average of 20 hours aweek on homework outside of the classroom, with a standard deviation of 4 hours per week.

If a random sample of 30 students from this community college is selected, find the probability (to 4 decimals) that thesample mean weekly homework hours will be:

a. at least 22 hours

b. between 18 and 22 hours

c. less than 10 hours per week

4. What is the minimum sample size needed for a 99% confidence interval estimate for the population proportion to have amaximum margin of error of 0.06

a. if there is a preliminary estimate of 0.80?

b. if there is no preliminary estimate, so the most conservative estimate must be used?

5. In a recent municipal survey, 2,000 randomly selected taxpayers were sampled and 1,200 adults stated that they are infavor of constructing a new hockey arena.

Construct a 90% confidence interval (calculated to 4 decimal places) to estimate the percentage of all municipal taxpayersthat are in favor of constructing the hockey arena.

6. Past market research indicates that the ages of all the regular customers of a large fitness club are normally distributed. Arecent sample of 6 randomly selected regular customers resulted in the following stem-and-leaf display of the ages of theselected customers:

Construct a 95% confidence interval estimate for the population mean age of all the club’s regular customers.

7. A medical researcher wishes to estimate, within 2 points, the average systolic blood pressure of university studentslocated in a Canadian province. If the researcher wishes to be 96% confident, how large a sample should she select if thepopulation standard deviation systolic blood pressure for all the provincial university students is 6.0?

8. A census survey indicates that the national average family size was 3.25 persons per family in 2015. A 2018 sample offamilies randomly selected across the country results in the following family sizes:

4, 2, 3, 2, 1, 3, 4, 2, 5, 4

Assuming that the population of family sizes is normally distributed, conduct a test of hypothesis at the 5% level todetermine if the average family size has decreased between 2015 and 2018.

a. Show all key steps using the approach.

b. Show all key steps using the critical value approach.

9. A large online retail company claims that more than 80% of all its orders are delivered to customers’ homes within72 hours. A researcher working for the Department of Consumer and Corporate Affairs, suspicious of this claim, took arandom sample of 400 orders and found that 330 of them were delivered to homes within a 72 hour period. Conduct atest of hypothesis at the 1% level to determine if the random sample supports the retailer’s claim.



10. In 2014 the average cost of all weddings in the country was $23,000. A recent sample of 64 couples who got married thisyear produced a mean wedding cost of $24,500 with a standard deviation of $4,400. Conduct a test of hypothesis atthe 5% level to determine if the average cost of weddings has changed.


b. How strong is the evidence against the null hypothesis ( )? Explain your reasoning. (See Additional Topic 4C:


21

The and Critical Value Approaches in Unit 4 of the Study Guide, Section 4-7.)

Mann, Prem S. Introductory Statistics, 9th ed. Wiley, 2016. [VitalSource].


22

Show all your work and keep your calculations to four decimals, unless otherwise stated. You can access the solutions to this self-

test on the course home page.


a. T F The standard deviation of the sampling distribution of the sample mean is equal to the populationstandard deviation.

b. T F If the population distribution is positively skewed, then the sampling distribution of the sample meanis also positively skewed.

c. T F When the population standard deviation is unknown and the sample size exceeds 30, the distribution is used to compute a confidence interval for the population mean.

d. T F When the population standard deviation is known and the sample size exceeds 30, the distribution isused to compute a confidence interval for the population mean.

e. T F A larger sample size will tend to reduce the width of a confidence interval.

f. T F In conducting a test of hypothesis, if the exceeds the level of significance, we reject the nullhypothesis.

g. T F In conducting a test of hypothesis, if the alternative hypothesis consists of a “ ” expression, thecritical value will be a negative number.

h. T F In conducting a test of hypothesis, if the is less than 0.001, the evidence against the nullhypothesis is considered to be very weak.







i. less than 0.30




If a random sample of 30 students from this community college is selected, find the probability (to 4 decimals) that the

MATH 215 C10 - Self-Test: Unit 4

1

sample mean weekly homework hours will be:













4, 2, 3, 2, 1, 3, 4, 2, 5, 4









b. How strong is the evidence against the null hypothesis ( )? Explain your reasoning. (See Additional Topic 4C:The and Critical Value Approaches in Unit 4 of the Study Guide, Section 4-7.)

MATH 215 C10 - Self-Test: Unit 4

2

Show all your work and keep your calculations to four decimal places, unless otherwise stated.


a. T The standard deviation of the sampling distribution of the sample mean is equal to the populationstandard deviation.

b. T If the population distribution is positively skewed, then the sampling distribution of the sample meanis also positively skewed.

c. T When the population standard deviation is unknown and the sample size exceeds 30, the distribution is used to compute a confidence interval for the population mean.

d. F When the population standard deviation is known and the sample size exceeds 30, the distribution isused to compute a confidence interval for the population mean.

e. F A larger sample size will tend to reduce the width of a confidence interval.

f. T In conducting a test of hypothesis, if the exceeds the level of significance, we reject the nullhypothesis.

g. F In conducting a test of hypothesis, if the alternative hypothesis consists of a “ ” expression, thecritical value will be a negative number.

h. T In conducting a test of hypothesis, if the is less than 0.001, the evidence against the nullhypothesis is considered to be very weak.




Answer: The sampling distribution would be a normal distribution according to the central limit theorem, asboth ( ) and ( ) exceed five.


Solution:


Solution:


F

F

F

T

T

F

T

F

MATH 215 C10 - Self-Test Answer Key: Unit 4

1

i. less than 0.30

Solution:


Solution:


Solution:


If a random sample of 30 students from this community college is selected, find the probability (to 4 decimals) that thesample mean weekly homework hours will be


Solution:


2


Solution:


Solution:



Solution:

Given , , ,

find such that 0.995 of the distribution is to the left of .


3


Solution:



Solution:

Find .

Find from Table IV (Appendix B in the eText), such that the area to the left of :

The confidence interval is:

.



4


Solution:

Formula to use: confidence interval:

To find the sample mean and standard deviation:

Note that the six ages in the sample are 18, 19, 22, 24, 26 and 33.

Sample mean

To find : degrees of freedom .

Find such that the area to the right of equals 0.025: .

Confidence interval:

Answer:


Solution:


5

Formula to use:

To find :

is such that 98% of the distribution falls to the left of . So, from the normal table, .

, so is the desired sample size.


4, 2, 3, 2, 1, 3, 4, 2, 5, 4



Solution:

Let = mean family size.


: (Mean family size in 2015)

: (Mean family size less than 3.25)


Select the distribution, as the population is normally distributed and the population standard deviation isunknown.


To find the , we must first find the sample test statistic “ ” as follows:

, where and .

To find and :

µ


6

The is the area to the left of the sample test statistic -0.6339 under the curve asshown below.

Steps in estimating the : Read down the in Table V in Appendix B of the eText until you findthe appropriate degrees of freedom, in this case . Ignoring the sign of the calculated teststatistic, locate 0.6339 in the row with 9 degrees of freedom ( ). It falls in front of (to left of) 1.383. Thisvalue corresponds to an area that exceeds 0.10. So, we can conclude that the exceeds 0.10.


7

Source: Adapted from Prem S. Mann, Introductory Statistics, 9th ed., B21.


Since the , which exceeds 0.10, exceeds the level of significance of 0.05, we do not reject . Wecannot conclude that the average family size has decreased below the 2015 value of 3.25 persons perfamily.


Solution:

Let = mean family size


: (mean family size in 2015)

: (mean family size less than 3.25)


Select the distribution, as the population is normally distributed and the population standard deviation isunknown.

Step 3: Determine the rejection and non-rejection regions.

Given that and that for , the area in the left tail is 0.05. The degrees of freedom are. The critical value is -1.833.



Since the sample test statistic -0.6339 is NOT in the rejection region, we do NOT reject the null hypothesis.We cannot conclude that the average family size has decreased below the 2015 value of 3.25 persons perfamily.


8



Solution:

Note: In this solution, there are two meanings for “ .” The is a probability. The in the hypothesisstatements means the proportion of orders delivered within the stated time period.

Let = proportion of orders delivered within 72 hours.

Step 1: State the null hypothesis ( ) and alternative hypothesis ( ).

:

: (proportion of delivered orders exceeds 80%)


We use the distribution, as both and exceed five.


To find the we must first find the sample test statistic as follows:

The is the area to the right of the sample test statistic 1.25 under the curve, whichequals as shown below.


Since the of 0.1056 exceeds the level of significance of 0.01, we do not reject he null hypothesis, so,we cannot conclude that more than 80% of all the company’s orders are delivered to customers’ homeswithin 72 hours.


Solution:

Let = proportion of orders delivered within 72 hours.


:

: (proportion of delivered orders exceeds 80%)


We use the distribution, as both and exceed five.

Step 3: Determine the rejection and non-rejection regions.

Given that and that is in , the critical value is 2.33 and the rejection and non-


9

rejection regions are:



Since the sample test statistic 1.25 is not beyond the critical value 2.33 and therefore, is in the non-rejection region, we do not reject the null hypothesis. We cannot conclude that more than 80% of all thecompany’s orders are delivered to customers’ homes within 72 hours.



Solution:

Let = average wedding cost


:

: (mean wedding cost has changed)


Select the distribution as the sample size is large and the population standard deviation is unknown.


To find the we must first find the sample test statistic “ ” as follows:

The is the combined area beyond the two values as shown below.


10

© Athabasca University

Steps in estimating the : Read down the in Appendix V of the eText until you find theappropriate degrees of freedom, which in this case is: . Ignoring the sign of the calculatedtest statistic, locate it (2.7273) in the row with 63 degrees of freedom. It falls in between 2.656 and 3.225.Read to the top of the table to locate the areas related to 2.656 and 3.225, which are 0.005 and 0.001.Since this is a two-tailed test, the estimated is between and

.


Since the estimated is between 0.002 and 0.01, which is less than the 0.05 level of significance, wereject . Therefore, we can conclude that the mean wedding cost has changed from its 2014 amount.

b. How strong is the evidence against the null hypothesis ( )? Explain your reasoning. (See Additional Topic 4C:The and Critical Value Approaches in Unit 4 of the Study Guide, Section 4-7.)

Answer: A ranging between 0.01 and 0.002 is classed as “strong evidence” against the null hypothesis.


11

MATH 215 C10 - Study Guide: Unit 4

Documents

Transcript of MATH 215 C10 - Study Guide: Unit 4