Lesson 03 chapter 6 sampling

Statistics 2Statistics 2

Dr. Ning DING

IBS I.007

[email protected]

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Table of Contents

Chapter 6 Sampling- Review: Sampling and Standard Error

- Calculating Standard Error-Infinite Population

- Calculating Stanard Error-Finite Population

Chapter 7 Introduction of Estimation- Types of Estimates

- Interval Estimates

SPSS Tips for t-test

Sampling and Sampling Distribution

PopulationPopulation

= all items chosen for study

SampleSample

= a portion chosen from the population

ParameterParameter StatisticStatistic

Greek or capital letters Lowercase Roman letters

Chapter 6 Sampling- Review:

Sampling and Standard Error



Chapter 7 Introduction of Estimation- Types of

Estimates



Sampling and Sampling Distribution

PopulationPopulation SampleSample

ParameterParameter StatisticStatistic

N = numberμ = meanσ = standard deviation

N = numberμ = meanσ = standard deviation

n = numberX = meanSD = standard deviation

n = numberX = meanSD = standard deviation






Estimates



Sampling Distribution

MeanMean MeanMean MeanMean






Estimates



= Standard deviation of the distribution of a sample statistic

Standard Error

LargerStandard Error

Smaller Standard Error

Which one is better?Which one is better?Which one is better?Which one is better?






Estimates



Standard ErrorChapter 6 Sampling- Review:





Estimates



Standard Error

Sample size Sample size

Dispersion of sample meansDispersion of sample means

Standard ErrorStandard Error






Estimates



Standard Error

µ = 100σ = 25

=95

=106

=101

Population Range=80~240Population Range=80~240

Sample Range=90~120Sample Range=90~120

Standard Error Standard Error of meanof mean

Standard Deviation Standard Deviation of populationof population________






Estimates








Estimates



Calculating the Standard Error

Sample means

freq

uen

cy


individual savings accounts µ= $2000σ= $600

Sample= 100 accounts

the probability that the sample mean lies betw. $1900~$2050 ?the probability that the sample mean lies betw. $1900~$2050 ?

Standard Error of the mean

Population standard deviation

Sample size

Example:Example:






Estimates




the probability that the sample mean lies betw. $1900~$2050 ?the probability that the sample mean lies betw. $1900~$2050 ?

Sample meanPopulation mean

Standard error of the mean

0.45250.4525

0.29670.2967++

= 0.7492= 0.749274.92% of our sample means lies betw. $1900~$205074.92% of our sample means lies betw. $1900~$2050






Estimates




6-30 6-30

Chapter 6, No. 6-30 P.321

Known: Normal distribution, μ=375 σ=48 P=95%n = ?






Estimates




6-30 6-30

Chapter 6, No. 6-30 P.321

Known: Normal distribution, μ=375 σ=48 P=95% n=?

Step 1: P=Pz1+Pz2=0.950

z1=-1.96 z2=1.96

370< <380

-1.96< z <1.96 Step 2: 1.96=

Step 3: n=354.04

The sample size is at least 355The sample size is at least 355






Estimates




Infinitepopulation

Infinitepopulation

Finite population

Finite population






Estimates



The Finite Population Multiplier

population size

sample size

Finite population multiplierFinite population multiplierF.P.M.F.P.M.






Estimates



The Finite Population Multiplier

1) N= 20 n=5 0.8880.888

2) N= 20 n=19 0.2290.229

3) N= 20 n=20 00

4) N= 1000 n=20 0.990.99

When to use F.P.M.? When to use F.P.M.? If If






Estimates




SC 6-7aSC 6-7a

Chapter 6, SC No. 6-7 P.327

Known: N=125 n=64 μ=105 σ=17 =?

Step 1: n/N=64/125= 0.512 >0.05 Yes, it is allowed to use F.P.M.

Step 2: =

==






Estimates




SC 6-7bSC 6-7b


Known: N=125 n=64 μ=105 σ=17 =1.4904

P(107.5<Xmean<109) = ?

Step 1: visualize and calculate z scores

==

P1=0.4535

P2=0.4963

P=0.4963-0.4535=0.0428






Estimates




SC 6-8SC 6-8


Known: n=36 μ=? σ=1.25 pounds

What is the probability that the sample mean is within one-half pound of the population mean?

==

Step 1: Visualize and calculate z scores

Step 2: Calculate the standard error of sample means






Estimates




SC 6-8SC 6-8


Known: n=36 μ=? σ=1.25 pounds

What is the probability that the sample mean is within one-half pound of the population mean?

==

Step 2: Calculate the standard error of sample means

Step 3: Calculate the z scores

Pz1=0.4918

Pz2=0.4918

++

= 0.9836= 0.9836

Step 4: convert to P value

Step 5: Finalize your answer






Estimates



Chapter 7 Introduction of Estimation

confidence levelconfidence level

confidence intervalconfidence interval






Estimates



Types of Estimates

Interval EstimatesInterval

EstimatesPoint

EstimatesPoint

Estimates






Estimates



Interval Estimates: Basic Concepts

P.354


Estimates

standard deviation is 10interviewed 200 personaccording to them, the mean is 36 months


Stardard error of the mean from an infinite population

standard deviation of the population

sample size

36+0.707=36.70736-0.707=35.293






Estimates




P.354


Estimates



z=1.0 P=0.3413

68.3% of the actual mean lie between 35.293 and 36.707







Estimates




P.354


Estimates



z=2.0 P=0.4775

95.5% of the actual mean lie between _______ and_________

95.5% of the actual mean lie between _______ and_________








Estimates




P.354


Estimates



z=3.0 P=0.4987








Estimates




P.354


EstimatesChapter 6 Sampling- Review:





Estimates





Estimates

Interval =






Estimates





Estimates

EX 7-27aEX 7-27a

Chapter 7, No. 7-27 P. 365

Known: n=40

=

EX 7-27bEX 7-27b P=90% z=1.645

Interval =

Upper limit =1416+7.8029

Lower limit=1416-7.8029

=1424

=1408

90% confident that our population mean lies between 1408 and 1424.

90% confident that our population mean lies between 1408 and 1424.


Estimates






Estimates





Estimates

Interval =

If the σ is unknown, ?

Interval =






Estimates



Estimated standard error of mean

Estimated standard error of proportionproportion

P.368

Interval =



Estimates

EX 7-35aEX 7-35a known: n=200 p=0.05 q=0.95

EX 7-35bEX 7-35b known: n=200 p=0.05 q=0.95 P=98% z=2.33

Interval =

Answer: 0.01 ~0.09Answer: 0.01 ~0.09






Estimates



Chapter 7, No. 7-35 P. 369



Estimates

If the sample size is =< 30,AND σ is unknown, ?

t - distributiont - distribution

You can read the t value from Appendix Table 2You can read the t value from Appendix Table 2






Estimates





Estimates

How to read the t-table ? t - distributiont - distribution

e.g. n=10 df=9 P=90%

0.05

0.05






Estimates





Estimates

How to use t value ? t - distributiont - distribution






Estimates



interval =

interval =

Summary

Chapter 6 Sampling- Review: Sampling and Standard Error



Chapter 7 Introduction of Estimation- Types of Estimates


The Normal DistributionSPSS Tip: t-test

The data can be downloaded from:

Blackboard – Inductive Statsitics STA2—SPSS--Week 3


3 types of t-test3 types of t-test

One Sample t-testOne Sample t-test Paired-Samples t-testPaired-Samples t-test Independent Samples t-testIndependent Samples t-test

test whether the population mean is different from a constant

test whether the population mean of differences between paired scores is equal to zero

test the relationship between two categories and a quantitative variable

The Normal DistributionSPSS Tip: t-test One Sample t-testOne Sample t-test

Variable DescriptionPCAS Price Change Attitude Scale

Example:A researcher wants to evaluate whether customers believe price change is more a function of natural fluctuations in inflation or due to effects caused by human interventions. Thirty customers are assessed on the Price Change Attitude Scale, which yields scores that range from 0 (due solely to natural fluctuations in inflation) to 100 (due solely to human interventions). A score of 50 is the test value and represents an equal contribution of the two effects.


Blackboard – Inductive Statsitics STA2—SPSS--Week 3 One-Sample t-test.sav


Variable DescriptionPCAS Price Change Attitude Scale

Example:A researcher wants to evaluate whether customers believe price change is more a function of natural fluctuations in inflation or due to effects caused by human interventions. Thirty customers are assessed on the Price Change Attitude Scale, which yields scores that range from 0 (due solely to natural fluctuations in inflation) to 100 (due solely to human interventions). A score of 50 is the test value and represents an equal contribution of the two effects.

Null Hypothesis: The population mean is equal to 50.


Step 1: Choose Analyze--> Compare Means --> One-Sample T TestStep 1: Choose Analyze--> Compare Means --> One-Sample T Test

One Sample t-testOne Sample t-test


Step 2: Move the variable you want to test into the box ”Test Variable(s)”. Enter the value in the box “Test Value”. In this example, the PCAS middle value is 50. Click OK and you will see a popup window.

Step 2: Move the variable you want to test into the box ”Test Variable(s)”. Enter the value in the box “Test Value”. In this example, the PCAS middle value is 50. Click OK and you will see a popup window.



Read the next slide to know how to interpret it !

The Normal DistributionSPSS Tip: t-test•The mean score on the PCAS is 62.30, which is 12.3000 (labeled mean difference) above the test value of 50. The standard deviation of the PCAS scores is 12.089. •We are able to reject the null hypothesis that population mean is equal to 50. We reject the null hypothesis because the significance value or p-value (displayed as .000) is less than the traditional alpha of .05. The p-value is associated with the t value of 5.573 with degrees of freedom of 29. •The 95 percent confidence interval of the difference between the mean PCAS and the test value ranges from 7.79 to 16.81. •An effect size statistic – the standardized mean difference – can be computed by dividing the mean difference by the standard deviation. For our example, it is equal to 1.017; that is, 12.300/12.089=1.017, a moderate value.



Variable DescriptionPre_SSC Percent correct on the Sales Scale for Customers prior to the

new TV commercialPost_SSC Percent correct on the Sales Scale for Customers after the

new TV commercial

Example:A researcher is interested in determining whether customers’ satisfaction with DOVE body lotion improves when exposed to a new TV commercial. Thirty customers are assessed on the Satisfaction Scale for Customers (SSC) prior to and after the new TV commercial.


Blackboard – Inductive Statsitics STA2—SPSS--Week 3 Paired-Sample t-test.sav

Paired-Samples t-testPaired-Samples t-test


Variable DescriptionPre_SSC Percent correct on the Sales Scale for Customers prior to the

new TV commercialPost_SSC Percent correct on the Sales Scale for Customers after the

new TV commercial

Example:A researcher is interested in determining whether customers’ satisfaction with DOVE body lotion improves when exposed to a new TV commercial. Thirty customers are assessed on the Satisfaction Scale for Customers (SSC) prior to and after the new TV commercial.

Paired-Samples t-testPaired-Samples t-test

Null Hypothesis: The population means’ difference is zero.

The Normal DistributionSPSS Tip: t-test Paired-Samples t-testPaired-Samples t-test

Step 1: Choose Analyze--> Compare Means --> Paired-Samples T TestStep 1: Choose Analyze--> Compare Means --> Paired-Samples T Test


Step 2: Move the first variable the box “Paired Variables”, to the location Variable 1, and the second variable to Variable 2. Click OK and you will see a popup window.

Step 2: Move the first variable the box “Paired Variables”, to the location Variable 1, and the second variable to Variable 2. Click OK and you will see a popup window.


Read the next slide to know how to interpret it !


•The mean and standard deviation of the Pre_SSC scores are 73.73 and 16.101, respectively. The mean and standard deviation of the Post_SSC scores are 76.50 and 13.117, respectively. •The mean and standard deviation of the paired differences between the Post SSC and Pre_SSC scores is 2.767 and 8.178, respectively. Note that the mean of the paired differences is equal to the difference in the means: 2.767= 76.50-73.73. •We are not able to reject the null hypothesis that population mean difference is equal to zero because the significance value or p-value of .074 is greater than the traditional alpha of .05. The p-value is associated with the t of 1.86=53 with degree of freedom of 29. •The 95 percent confidence interval of the mean of the paired differences in SSC scores ranges from -.287 to 5.820. •An effect size statistic – the standardized mean difference – can be computed by dividing the mean of the paired differences by the standard deviation of the paired differences. For our example, it is equal to .34; that is, 2.767/8.178 = .34.

Lesson 03 chapter 6 sampling

Education

Transcript of Lesson 03 chapter 6 sampling