Margin of Error and the Interval Estimate Interval ...

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WEEK 7:SAMPLING DISTRIBUTIONS

Dr. Po-Lin Lai

1 Interval Estimation

Population Mean: Known Population Mean: Unknown Determining the Sample Size Population Proportion

Margin of Error and the Interval Estimate3

A point estimator cannot be expected to provide theexact value of the population parameter.

An interval estimate can be computed by adding andsubtracting a margin of error to the point estimate.

Point Estimate +/ Margin of Error

The purpose of an interval estimate is to provideinformation about how close the point estimate is tothe value of the parameter.

Margin of Error and the Interval Estimate4

The general form of an interval estimate of apopulation mean is

Interval Estimation

Provides a range of values• Based on observations from one sample

Gives information about closeness to unknown population parameter

• Stated in terms of probability– Knowing exact closeness requires knowing unknown

population parameter

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Interval Estimate of a Population Mean: Known6

In order to develop an interval estimate of a population mean, the margin of error must be computed using either:• the population standard deviation , or• the sample standard deviation s

is rarely known exactly, but often a good estimate can be obtained based on historical data or other information.

We refer to such cases as the known case.

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Estimation Process: Known

Mean, , is unknown

Population

Sample

Random Sample

I am 95% confident that is between 40 &

60.

Meanx = 50

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Key Elements of Interval Estimation

Sample statistic (point estimate)

Confidence interval

Confidence limit (lower)

Confidence limit (upper)

A confidence interval provides a range of plausible values for the population parameter.

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According to the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal for large samples. Let us calculate the interval estimator:

Confidence Interval

That is, we form an interval from 1.96 standard deviations below the sample mean to 1.96 standard deviations above the mean. Prior to drawing the sample, what are the chances that this interval will enclose µ, the population mean?

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If sample measurements yield a value of that falls between the two lines on either side of µ, then the interval will contain µ.

Confidence Interval

The area under the normal curve between these two boundaries is exactly .95. Thus, the probability that a randomly selected interval will contain µ is equal to .95.

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The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter - that is, the relative frequency with which similarly constructed intervals enclose the population parameter when the estimator is used repeatedly a very large number of times. The confidence level is the confidence coefficient expressed as a percentage.

Confidence Coefficient11

If our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain µ and 5% will not.

95% Confidence Level

For a confidence coefficient of 95%, the area in the two tails is .05. To choose a different confidence coefficient we increase or decrease the area (call it ) assigned to the tails. If we place /2 in each tail and z/2 is the z-value, the confidence interval with coefficient coefficient (1 – ) is

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3

There is a 1 probability that the value of asample mean will provide a margin of error of or less.

/2/2 /2/21 - of allvalues

1 - of allvalues

Samplingdistribution

of


of

Interval Estimate of a Population Mean: Known Interval Estimate of a Population Mean: Known

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/2/2 /2/21 - of allvalues

1 - of allvalues


of


of

[------------------------- -------------------------][------------------------- -------------------------]

[------------------------- -------------------------]

intervaldoes notinclude interval

includes interval

includes


Interval Estimate of

where: is the sample mean1 - is the confidence coefficientz/2 is the z value providing an area of

/2 in the upper tail of the standard normal probability distribution

is the population standard deviationn is the sample size


Values of z/2 for the Most Commonly Used Confidence Levels

90% .10 .05 .9500 1.645

Confidence TableLevel /2 Look-up Area z/2

95% .05 .025 .9750 1.96099% .01 .005 .9950 2.576

Meaning of Confidence17

Because 90% of all the intervals constructed usingwill contain the population mean,

we say we are 90% confident that the interval includes the population mean .

Because 90% of all the intervals constructed usingwill contain the population mean,

we say we are 90% confident that the interval includes the population mean .

We say that this interval has been established at the90% confidence level.

The value .90 is referred to as the confidence coefficient.

Thinking Challenge

You’re a Q/C inspector for Gallo. The for 2-liter bottles is .05 liters. A random sample of 100 bottles showed x = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles?

2 liter

© 1984-1994 T/Maker Co.

2 liter

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Confidence Interval Solution*19


Example: Discount SoundsDiscount Sounds has 260 retail outlets throughout the United States. The firm is evaluating a potential location for a new outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location.A sample of size n = 36 was taken; the sample mean income is $41,100. The population is not believed to be highly skewed. The population standard deviation is estimated to be $4,500, and the confidence coefficient to be used in the interval estimate is .95.


Example: Discount Sounds

95% of the sample means that can be observedare within + 1.96 of the population mean .

The margin of error is:

Thus, at 95% confidence, the margin of erroris $1,470.


Example: Discount SoundsInterval estimate of is:

We are 95% confident that the interval contains thepopulation mean.

$41,100 + $1,470or

$39,630 to $42,570


Adequate Sample Size

In most applications, a sample size of n = 30 is adequate.

If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is recommended.

If the population is believed to be at least approximatelynormal, a sample size of less than 15 can be used.

If the population is not normally distributed but is roughlysymmetric, a sample size as small as 15 will suffice.



If the population is believed to be at least approximatelynormal, a sample size of less than 15 can be used.

If the population is not normally distributed but is roughlysymmetric, a sample size as small as 15 will suffice.

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Interval Estimate of a Population Mean: Unknown25

If an estimate of the population standard deviation cannot be developed prior to sampling, we use the sample standard deviation s to estimate .

This is the unknown case. In this case, the interval estimate for is based on

the t distribution. (We’ll assume for now that the population is normally

distributed.)

t Distribution26

William Gosset, writing under the name “Student”,is the founder of the t distribution.

Gosset was an Oxford graduate in mathematics andworked for the Guinness Brewery in Dublin.

He developed the t distribution while working onsmall-scale materials and temperature experiments.

The t distribution is a family of similar probability distributions.

t Distribution27

A specific t distribution depends on a parameter known as thedegrees of freedom.

Degrees of freedom refer to the number of independent pieces of information that go into the computation of s.

A t distribution with more degrees of freedom has less dispersion.

As the degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller.

t Distribution

Standardnormal

distribution

t distribution(20 degreesof freedom)

t distribution(10 degreesof freedom)

0z, t

t Distribution29

For more than 100 degrees of freedom, the standardnormal z value provides a good approximation tothe t value.

The standard normal z values can be found in theinfinite degrees ( ) row of the t distribution table.The standard normal z values can be found in theinfinite degrees ( ) row of the t distribution table.

Degrees of Freedom

The actual amount of variability in the sampling distribution of t depends on the sample size n.A convenient way of expressing this dependence is to say that the t-statistic has (n – 1) degrees of freedom (df).

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t Distribution31

Standard normalz values

Interval Estimate of a Population Mean: Unknown

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Interval Estimate

where: 1 - = the confidence coefficientt/2 = the t value providing an area of /2

in the upper tail of a t distributionwith n - 1 degrees of freedom

s = the sample standard deviation


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Example: Apartment RentsA reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 16 efficiency apartments within a half-mile of campus resulted in a sample mean of $750 per month and a sample standard deviation of $55.Let us provide a 95% confidence interval estimate of the mean rent per month for the population of efficiency apartments within a half-mile of campus. We will assume this population to be normally distributed.


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At 95% confidence, 　 = .05, and 　/2 = .025.

t.025 is based on n - 1 = 16 - 1 = 15 degrees of freedom. In the t distribution table we see that t.025 = 2.131.


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Interval Estimate

We are 95% confident that the mean rent per monthfor the population of efficiency apartments within ahalf-mile of campus is between $720.70 and $779.30.

Marginof Error

Thinking ChallengeYou’re a time study analyst in manufacturing. You’ve recorded the 6 task times (min.): mean= 3.7 and S.D. =3.8987. What is the 90% confidence interval estimate of the population mean task time?

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Confidence Interval Solution* x = 3.7

s = 3.8987

• n = 6, df = n – 1 = 6 – 1 = 5

• t.05 = 2.015

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If the population distribution is highly skewed orcontains outliers, a sample size of 50 or more isrecommended.

In most applications, a sample size of n = 30 isadequate when using the expression todevelop an interval estimate of a population mean.

In most applications, a sample size of n = 30 isadequate when using the expression todevelop an interval estimate of a population mean.


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If the population is believed to be at leastapproximately normal, a sample size of less than 15can be used.

If the population is not normally distributed but isroughly symmetric, a sample size as small as 15 will suffice.

Summary of Interval Estimation Proceduresfor a Population Mean

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Can thepopulation standard

deviation be assumed known ?

Use

Yes No

Use

KnownCase

UnknownCase

Use the samplestandard deviation

s to estimate

Sample Size for an Interval Estimate of a Population Mean

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Let E = the desired margin of error.

E is the amount added to and subtracted from thepoint estimate to obtain an interval estimate.

If a desired margin of error is selected prior tosampling, the sample size necessary to satisfy themargin of error can be determined.

Sample Size for an Interval Estimateof a Population Mean

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Margin of Error

Necessary Sample Size

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The Necessary Sample Size equation requires avalue for the population standard deviation .

If is unknown, a preliminary or planning valuefor can be used in the equation.

1. Use the estimate of the population standarddeviation computed in a previous study.

2. Use a pilot study to select a preliminary study anduse the sample standard deviation from the study.

3. Use judgment or a “best guess” for the value of .


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Example: Discount SoundsRecall that Discount Sounds is evaluating a potential location for a new retail outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location.Suppose that Discount Sounds’ management team wants an estimate of the population mean such that there is a 0.95 probability that the sampling error is $500 or less.How large a sample size is needed to meet the required precision?


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At 95% confidence, z.025 = 1.96. Recall that 　　= 4,500.

A sample of size 312 is needed to reach a desiredprecision of + $500 at 95% confidence.

Interval Estimateof a Population Proportion

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The general form of an interval estimate of apopulation proportion is

The sampling distribution of plays a key role incomputing the margin of error for this intervalestimate.

The sampling distribution of plays a key role incomputing the margin of error for this intervalestimate.

The sampling distribution of can be approximatedby a normal distribution whenever np > 5 andn(1 – p) > 5.

The sampling distribution of can be approximatedby a normal distribution whenever np > 5 andn(1 – p) > 5.


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/2/2 /2/2

Normal Approximation of Sampling Distribution of


of


of

pp

1 - of allvalues

1 - of allvalues


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Interval Estimate

where: 1 - is the confidence coefficientz/2 is the z value providing an area of

/2 in the upper tail of the standardnormal probability distributionis the sample proportion

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Example: Political Science, Inc.Political Science, Inc. (PSI) specializes in voter pollsand surveys designed to keep political office seekers informed of their position in a race.Using telephone surveys, PSI interviewers ask registered voters who they would vote for if the election were held that day. In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, favor a particular candidate. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of registered voters that favor the Candidate.


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where: n = 500, = 220/500 = .44, z/2 = 1.96

PSI is 95% confident that the proportion of all votersthat favor the candidate is between .3965 and .4835.PSI is 95% confident that the proportion of all votersthat favor the candidate is between .3965 and .4835.

= .44 + .0435

Sample Size for an Interval Estimateof a Population Proportion

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Margin of Error

Solving for the necessary sample size, we get

However, will not be known until after we have selected the sample. We will use the planning valuep* for .


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Necessary Sample Size

The planning value p* can be chosen by:1. Using the sample proportion from a previous sample

of the same or similar units, or2. Selecting a preliminary sample and using the

sample proportion from this sample.3. Use judgment or a “best guess” for a p* value.4. Otherwise, use .50 as the p* value.


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Example: Political Science, Inc.Suppose that PSI would like a .99 probability that the sample proportion is within + .03 of the population proportion.How large a sample size is needed to meet the required precision? (A previous sample of similar units yielded .44 for the sample proportion.)


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At 99% confidence, z.005 = 2.576. Recall that = .44.

A sample of size 1817 is needed to reach a desiredprecision of + .03 at 99% confidence.

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NoteWe used .44 as the best estimate of p in the preceding expression. If no information is available about p, then .5 is often assumed because it provides the highest possible sample size. If we had used p = .5, the recommended nwould have been 1843.

Hypothesis Testing56

Developing Null and Alternative Hypotheses Type I and Type II Errors Population Mean: Known Population Mean: Unknown Population Proportion

Hypothesis Testing57

Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected.

The null hypothesis, denoted by H0 , is a tentative assumption about a population parameter.

The alternative hypothesis, denoted by Ha, is the opposite of what is stated in the null hypothesis.

The hypothesis testing procedure uses data from a sample to test the two competing statements indicated by H0 and Ha.

Developing Null and Alternative Hypotheses It is not always obvious how the null and alternative

hypotheses should be formulated. Care must be taken to structure the hypotheses appropriately

so that the test conclusion provides the information the researcher wants.

The context of the situation is very important in determining how the hypotheses should be stated.

In some cases it is easier to identify the alternative hypothesis first. In other cases the null is easier.

Correct hypothesis formulation will take practice.

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Alternative Hypothesis as a Research Hypothesis

Developing Null and Alternative Hypotheses

• Many applications of hypothesis testing involve an attemptto gather evidence in support of a research hypothesis.

• In such cases, it is often best to begin with the alternativehypothesis and make it the conclusion that the researcher hopes to support.

• The conclusion that the research hypothesis is true is madeif the sample data provide sufficient evidence to show thatthe null hypothesis can be rejected.

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• Example: A new teaching method is developed that is believed to be better than the current method.

• Alternative Hypothesis: The new teaching method is better.

• Null Hypothesis: The new method is no better than the old method.

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• Example: A new sales force bonus plan is developed in an

attempt to increase sales.

• Alternative Hypothesis: The new bonus plan increase sales.

• Null Hypothesis: The new bonus plan does not increase sales.

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• Example: A new drug is developed with the goal of lowering blood pressure more than the existing drug.

• Alternative Hypothesis: The new drug lowers blood pressure more thanthe existing drug.

• Null Hypothesis: The new drug does not lower blood pressure morethan the existing drug.

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Null Hypothesis as an Assumption to be Challenged

• We might begin with a belief or assumption thata statement about the value of a populationparameter is true.

• We then using a hypothesis test to challenge theassumption and determine if there is statisticalevidence to conclude that the assumption isincorrect.

• In these situations, it is helpful to develop the nullhypothesis first.

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• Example: The label on a soft drink bottle states that itcontains 67.6 fluid ounces.

• Null Hypothesis: The label is correct. > 67.6 ounces.

• Alternative Hypothesis: The label is incorrect. < 67.6 ounces.

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Summary of Forms for Null and Alternative Hypotheses about a Population Mean

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The equality part of the hypotheses always appears in the null hypothesis.

In general, a hypothesis test about the value of a population mean must take one of the following three forms (where 0is the hypothesized value of the population mean).

One-tailed(lower-tail)

One-tailed(upper-tail)

Two-tailed

Example: Metro EMS

Null and Alternative Hypotheses

A major west coast city provides one of the most comprehensive emergency medical services in the world. Operating in a multiple hospital system with approximately 20 mobile medical units, the service goal is to respond to medical emergencies with a mean time of 12 minutes or less.

The director of medical services wants to formulate a hypothesis test that could use a sample of emergency response times to determine whether or not the service goal of 12 minutes or less is being achieved.

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Example: Metro EMS

Null and Alternative Hypotheses67

The emergency service is meeting the response goal; no follow-up action is necessary.

The emergency service is not meeting the response goal; appropriate follow-up action is necessary.

H0:

Ha:

where: = mean response time for the populationof medical emergency requests

Type I Error

Because hypothesis tests are based on sample data,we must allow for the possibility of errors.

A Type I error is rejecting H0 when it is true.

The probability of making a Type I error when thenull hypothesis is true as an equality is called thelevel of significance.

Applications of hypothesis testing that only controlthe Type I error are often called significance tests.

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Type II Error

A Type II error is accepting H0 when it is false.

It is difficult to control for the probability of makinga Type II error.

Statisticians avoid the risk of making a Type IIerror by using “do not reject H0” and not “accept H0”.

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Type I and Type II Error70

CorrectDecision Type II Error

CorrectDecisionType I ErrorReject H0

(Conclude > 12)

Accept H0(Conclude < 12)

H0 True( < 12)

H0 False( > 12)Conclusion

Population Condition

p-Value Approach to One-Tailed Hypothesis Testing

Reject H0 if the p-value < .

The p-value is the probability, computed using thetest statistic, that measures the support (or lack ofsupport) provided by the sample for the nullhypothesis.

If the p-value is less than or equal to the level ofsignificance , the value of the test statistic is in therejection region.

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Lower-Tailed Test About a Population Mean: Known

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p-Value Approach p-Value Approach

p-valuep-value

00-z =-1.28-z =-1.28

= .10 = .10

zz

z =-1.46z =

-1.46

Samplingdistributionof


p-Value < ,so reject H0.

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Upper-Tailed Test About a Population Mean: Known

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p-Value Approach p-Value Approach

p-Valuep-Value

00 z =1.75z =1.75

= .04 = .04

zzz =2.29z =2.29



p-Value < ,so reject H0.

Critical Value Approach to One-Tailed Hypothesis Testing

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The test statistic z has a standard normal probabilitydistribution.

We can use the standard normal probabilitydistribution table to find the z-value with an areaof in the lower (or upper) tail of the distribution.

The value of the test statistic that established theboundary of the rejection region is called thecritical value for the test.

The rejection rule is:• Lower tail: Reject H0 if z < -z• Upper tail: Reject H0 if z > z

Lower-Tailed Test About a Population Mean: Known

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00z = 1.28z = 1.28

Reject H0Reject H0

Do Not Reject H0Do Not Reject H0

z



Critical Value Approach Critical Value Approach

Upper-Tailed Test About a Population Mean: Known

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00 z = 1.645z = 1.645

Reject H0Reject H0


z



Critical Value Approach Critical Value Approach

Steps of Hypothesis Testing77

Step 1. Develop the null and alternative hypotheses.Step 2. Specify the level of significance .Step 3. Collect the sample data and compute the test

statistic.

p-Value Approach

Step 4. Use the value of the test statistic to compute thep-value.

Step 5. Reject H0 if p-value < .

Steps of Hypothesis Testing78

Critical Value ApproachStep 4. Use the level of significance to determine the

critical value and the rejection rule.

Step 5. Use the value of the test statistic and the rejectionrule to determine whether to reject H0.

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One-Tailed Tests About a Population Mean: Known

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Example: Metro EMSThe response times for a random sample of 40 medical emergencies were tabulated. The sample mean is 13.25 minutes. The population standard deviation is believed to be 3.2 minutes.The EMS director wants to perform a hypothesis test, with a .05 level of significance, to determine whether the service goal of 12 minutes or less is being achieved.


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p -Value and Critical Value Approaches

1. Develop the hypotheses.

2. Specify the level of significance. = .05

H0: Ha:

3. Compute the value of the test statistic.


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p –Value Approaches

5. Determine whether to reject H0.

4. Compute the p –value.

For z = 2.47, cumulative probability = .9932.p–value = 1 .9932 = .0068

Because p–value = .0068 < = .05, we reject H0.

There is sufficient statistical evidenceto infer that Metro EMS is not meeting

the response goal of 12 minutes.


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p –Value Approaches

p-valuep-value

00 z =1.645z =1.645

= .05 = .05

zz

z =2.47z =2.47




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Critical Value Approaches


There is sufficient statistical evidenceto infer that Metro EMS is not meeting

the response goal of 12 minutes.

Because 2.47 > 1.645, we reject H0.

For = .05, z.05 = 1.645

4. Determine the critical value and rejection rule.

Reject H0 if z > 1.645

p-Value Approach toTwo-Tailed Hypothesis Testing

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The rejection rule:Reject H0 if the p-value < .

Compute the p-value using the following three steps:

3. Double the tail area obtained in step 2 to obtainthe p –value.

2. If z is in the upper tail (z > 0), find the area underthe standard normal curve to the right of z.

If z is in the lower tail (z < 0), find the area underthe standard normal curve to the left of z.

1. Compute the value of the test statistic z.

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Critical Approach toTwo-Tailed Hypothesis Testing

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The critical values will occur in both the lower andupper tails of the standard normal curve.

The rejection rule is:Reject H0 if z < -z/2 or z > z/2.

Use the standard normal probability distributiontable to find z/2 (the z-value with an area of /2 inthe upper tail of the distribution).

Two-Tailed Tests About a Population Mean: Known

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Example: Glow Toothpaste The production line for Glow toothpaste is designed to fill tubes with a mean weight of 6 oz. Periodically, a sample of 30 tubes will be selected in order to check the filling process.Quality assurance procedures call for the continuation of the filling process if the sample results are consistent with the assumption that the mean filling weight for the population of toothpaste tubes is 6 oz.; otherwise the process will be adjusted.


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Example: Glow ToothpasteAssume that a sample of 30 toothpaste tubes provides a sample mean of 6.1 oz. The population standard deviation is believed to be 0.2 oz.Perform a hypothesis test, at the .03 level of significance, to help determine whether the filling process should continue operating or be stopped and corrected.


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Example: Glow Toothpaste

1. Determine the hypotheses.

2. Specify the level of significance.


= .03

p –Value and Critical Value Approaches

H0: Ha:


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For z = 2.74, cumulative probability = .9969p–value = 2(1 .9969) = .0062

Because p–value = .0062 < = .03, we reject H0.There is sufficient statistical evidence to

infer that the alternative hypothesis is true(i.e. the mean filling weight is not 6 ounces).


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/2 =.015/2 =

.015

00z/2 = 2.17z/2 = 2.17

zz

/2 =.015/2 =

.015

p-Value Approach

-z/2 = -2.17-z/2 = -2.17z = 2.74z = 2.74z = -2.74z = -2.74

1/2p -value= .0031

1/2p -value= .0031

1/2p -value= .0031

1/2p -value= .0031

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Example: Glow ToothpasteCritical Value Approach


There is sufficient statistical evidence toinfer that the alternative hypothesis is true(i.e. the mean filling weight is not 6 ounces).


For /2 = .03/2 = .015, z.015 = 2.17


Reject H0 if z < -2.17 or z > 2.17


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/2 = .015/2 = .015

00 2.172.17

Reject H0Reject H0Do Not Reject H0Do Not Reject H0

zz

Reject H0Reject H0

-2.17-2.17

Critical Value Approach



/2 = .015/2 = .015

Confidence Interval Approach toTwo-Tailed Tests About a Population Mean

Select a simple random sample from the populationand use the value of the sample mean to developthe confidence interval for the population mean .(Confidence intervals are covered in Chapter 8.)

If the confidence interval contains the hypothesizedvalue 0, do not reject H0. Otherwise, reject H0.(Actually, H0 should be rejected if 0 happens to beequal to one of the end points of the confidenceinterval.)

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The 97% confidence interval for is

Confidence Interval Approach toTwo-Tailed Tests About a Population Mean

Because the hypothesized value for thepopulation mean, 0 = 6, is not in this interval,the hypothesis-testing conclusion is that thenull hypothesis, H0: = 6, can be rejected.

or 6.02076 to 6.17924

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Test Statistic

Tests About a Population Mean: Unknown

This test statistic has a t distributionwith n - 1 degrees of freedom.

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Tests About a Population Mean: Unknown

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Rejection Rule: p -Value Approach

H0: Reject H0 if t > t

Reject H0 if t < -t

Reject H0 if t < - t or t > t

H0:

H0:

Rejection Rule: Critical Value Approach

Reject H0 if p –value <

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p -Values and the t Distribution

The format of the t distribution table provided in moststatistics textbooks does not have sufficient detailto determine the exact p-value for a hypothesis test.

However, we can still use the t distribution table toidentify a range for the p-value.

An advantage of computer software packages is thatthe computer output will provide the p-value for thet distribution.

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A State Highway Patrol periodically samples vehicle speeds at various locations on a particular roadway. The sample of vehicle speeds is used to test the hypothesis H0: < 65.

Example: Highway Patrol

One-Tailed Test About a Population Mean: Unknown

The locations where H0 is rejected are deemed the best locations for radar traps. At Location F, a sample of 64 vehicles shows a mean speed of 66.2 mph with a standard deviation of 4.2 mph. Use = .05 to test the hypothesis.

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1. Determine the hypotheses.

2. Specify the level of significance.


= .05

p –Value and Critical Value Approaches

H0: < 65Ha: > 65


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p –Value Approach



For t = 2.286, the p–value must be less than .025(for t = 1.998) and greater than .01 (for t = 2.387).

.01 < p–value < .025

Because p–value < = .05, we reject H0.We are at least 95% confident that the mean speedof vehicles at Location F is greater than 65 mph.


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Critical Value Approach


We are at least 95% confident that the mean speed of vehicles at Location F is greater than 65 mph. Location F is a good candidate for a radar trap.


For = .05 and d.f. = 64 – 1 = 63, t.05 = 1.669


Reject H0 if t > 1.669


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00 t =1.669t =1.669

Reject H0Reject H0


t

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End of Week 7

Margin of Error and the Interval Estimate Interval ...

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