Copyright © 2013 Pearson Education, Inc. All rights reserved Chapter 9 Inferring Population Means.

Copyright © 2013 Pearson Education, Inc. All rights reserved

Chapter 9

Inferring Population

Means

1 - 2 Copyright © 2013 Pearson Education, Inc.. All rights reserved.

Learning Objectives

Understand when the Central Limit Theorem for sample means applies and know how to use it to find approximate probabilities for sample means.

Know how to test hypotheses concerning a population mean and concerning the comparison of two population means.

Understand how to find, interpret, and use confidence intervals for a single population mean and for the difference of two population means.

1 - 3 Copyright © 2013 Pearson Education, Inc.. All rights reserved.

Learning Objectives Continued

Understand the meaning of the p-value and of significance levels.

Understand how to use a confidence interval to carry out a two tailed hypothesis test for a population mean or for a difference of two population means.


9.1

Sample Means of Random Samples

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Statistics, Parameters, Means and Proportions

Mean and Standard Deviation if the survey question has a numerical variable.

Proportion if the survey question is Yes/No The confidence interval and hypothesis test always

refer to the population not the sample

Copyright © 2013 Pearson Education, Inc.. All rights reserved.

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Accuracy of the Sample Mean

If the sample mean is accurate, then the average of all sample means will equal the population mean.

If Simple Random Sampling is used the sample mean is accurate, also called unbiased.

Other sampling techniques to be looked at later produce results that are close to being unbiased.


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Precision and the Sample Mean

The precision of the sample mean describes how much variability there is from one sample mean to the next.

If the population standard deviation is small the sample mean will have more precision.

If the sample size is large the sample mean will have more precision.


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Simulating Many Sample Means

As the sample size increases Better Precision Accuracy Does Not Change


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Standard Error

The Standard Error is the standard deviation of the sampling distribution.


xn

x

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Standard Error and Sample Size

The Standard Error is smaller for larger sample sizes.

Increasing the sample size by a factor of 4 decreases the standard error by a factor of 2.

Increasing the sample size by a factor of 100 decreases the standard error by a factor of 10.


xn

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The mean cost per item at a grocery store is $2.75 and the standard deviation is $1.26. A shopper randomly puts 36 items in her cart. Is 2.75 a parameter or a statistic?

Parameter Predict the average cost per item in the shopper’s

cart. $2.75

Find the standard error for carts with 36 items.


1.260.21

36x

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Comparing Standard Errors

The mean income for residents of the city is $47,000 and the standard deviation is $12,000. Find the standard error for the following sample sizes


n = 1 n = 4 n = 16 n = 100

→ $12,000→ $6,000→ $3,000→ $1,200


9.2

The Central Limit Theorem for Sample

Means

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Conditions for the Central Limit Theorem for Sample Means

Random Sampling Technique One or Both of the Following:

Population is Normally Distributed Sample Size is Large

Population Size is At Least 10 Times Bigger Than the Sample Size


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What is a Large Enough Sample Size?

If the population distribution is not too far from Normal then the sample size can be small.

For most population distributions n = 25 or higher gives sufficient accuracy.

If the population distribution is far from normal, a larger sample size is needed.


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Central Limit Theorem For Means

Central Limit Theorem: If the conditions are met and the population has mean m and standard deviation s, then the sampling distribution will be approximately normal.


,Nn

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Visualizing the Central Limit Theorem

Population Distribution Skewed Right

Sampling Distribution Approximately Normal


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Applying the Central Limit Theorem

The distribution of women’s pulse rates is skewed right with m = 74 bpm, s = 13 bpm.

If 30 women are selected, find


( 72)P x

1374, (74,2.1)

38N N

( 72) 0.17P x

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Applying the Central Limit Theorem

The distribution of women’s pulse rates is skewed right with m = 74 bpm, s = 13 bpm.

If one woman is selected can you find

P(x < 72) No, since the distribution is skewed right, it is

not normal. Without more information about the distribution this probability cannot be found.


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Population, Sample, and Sampling Distributions The population distribution is the distribution of all

individuals that exist. The distribution of the sample is the distribution of

the individuals that were surveyed. The mean, standard deviation, and the shape are likely to

be close to the population distribution. The sampling distribution is the distribution of all

possible sample means of sample size n. The mean will be the same as the population mean, but

the shape will be approximately normal and the standard deviation will be smaller.


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The t-Distribution

If s is unknown, we cannot find the z-score. Use the sample standard deviation s instead.

is an estimate for the standard error


xt

sn

EST

sSE

n

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Facts About the t-Distribution

Bell shaped Tails a little bigger than Normal Given n there are n – 1 degrees of freedom. For large degrees of freedom, the distribution

is almost normal.Copyright © 2013 Pearson Education, Inc.. All rights reserved.


9.3

Answering Questions about the Mean of a

Population

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Confidence Interval for a Population Mean

Gives a plausible range of values for the population mean.

Confidence level gives the percent of all possible confidence intervals that contain the population mean.

Similar to confidence interval for a population proportion, but used for a quantitative variable.


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CI Example:

45 randomly selected college students worked on homework for an average of 9 hours per week. Their standard deviation was 2 hours. Find a 90% confidence interval for the population mean.

d.f. = 44 → t = 1.68,

Lower Bound: 9 – 1.68 x 0.30 ≈ 8.5 Upper Bound: 9 + 1.68 x 0.30 ≈ 9.5 (8.5,9.5)


ESTx t SE

20.30

45EST

sSE

n

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CI Interpretations: (8.5,9.5)


Interpretation of Confidence Interval: We are 90% confident that the population mean number of hours worked on homework for all college students is between 8.5 and 9.5 hours.


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CI Interpretations: (8.5,9.5)


Interpretation of Confidence Level: If many groups of 45 randomly selected students were surveyed, each survey would result in a different confidence interval. 90% of these confidence intervals will succeed in containing the actual population mean number of hours worked on homework and 10% will not contain the true population mean.


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Hypothesis Test for a Population Mean

The same four steps apply for a hypothesis test for a population mean:1. Hypothesize: State H0 and Ha.

2. Prepare: Choose a, check conditions and assumptions and determine the test statistic to use.

3. Compute to Compare: Compute the test statistic and the p-value and compare p with a.

4. Interpret: Reject or fail to Reject H0? Write down the conclusion in the context of the study.


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Hypothesis Test Example (by Formula)

Ford claims that its 2012 Focus gets 40 mpg on the highway. Does your Focus’ mpg differ from 40 mpg? You chart your Focus over 35 randomly selected highway trips and find it got 39.5 mpg with a standard deviation of 1.4 mpg.

1. Hypothesize H0: m = 40, Ha: m ≠ 40

2. Prepare Choose a = 0.05, Use t-statistic: random and

large sample


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Ford claims that it’s 2012 Focus gets 40 mpg on the highway. Does your Focus’ mpg differ from 40 mpg? You chart your Focus over 35 randomly selected highway trips and find it got 39.2 mpg with a standard deviation of 1.4 mpg.

3. Compute to Prepare

4. Interpret p-value = 0.04 < a = 0.05 Reject H0. Accept Ha. There is statistically

significant evidence to conclude that your Focus does not get 40 mpg on average.


39.5 402.11

1.435

t

value 0.04p


9.4

Comparing Two Population Means

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Independent vs. Dependent (Paired)

Two samples are dependent or paired if each observation from one group is coupled with a particular observation from the other group. Before and After Identical Twins Husband and Wife Older Sibling and Younger Sibling

If there is no pairing then the samples are independent.Copyright © 2013 Pearson Education, Inc.. All rights reserved.

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Independent (Ind) or Dependent (Dep)? Do women perform better on average than

men on their statistics final? 60 women and 40 men were surveyed.

40 people’s blood pressure was measured before and after giving a public speech. Does blood pressure change on average?

Is the average tip percent greater for dinner than lunch? 35 wait staff who worked both lunch and dinner looked at their receipts.

Are Americans more stressed out on average compared to the French? 50 from each country were given a stress test.


→ Ind

→ Dep

→ Dep

→ Ind

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Independent Samples Standard Error and Margin of Error

Degrees of Freedom is approximately the

smaller of n1 – 1 and n2 – 1. Use a computer or calculator for better

accuracy.


2 21 2

1 2EST

s sSE

n n

Margin of Error ESTt SE

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Requirement for Independent Samples

Both samples are randomly taken and each observation is independent of any other.

The two samples are independent of each other (not paired).

Either both populations are Normally distributed or each sample size is greater than 25.


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Example: Independent Samples

38 randomly selected engineer majors and 42 randomly selected psychology majors were observed to estimate the difference in how long it takes to graduate.Find a 95% confidence interval for the difference.

The two population are independent since there is no pairing between each engineer major and each psychology major.

The students were selected randomly, independently, and the sample sizes are both greater than 25.Copyright © 2013 Pearson Education, Inc.. All rights reserved.

5.1, 0.4, 5.6, 0.5E E P Px s x s

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38 randomly selected engineer majors and 42 randomly selected psychology majors were observed to estimate the difference in how long it takes to graduate.

Find a 95% confidence interval for the difference. Stat → T Statistics → Two sample → with summary


5.1, 0.4, 5.6, 0.5E E P Px s x s

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38 randomly selected engineer majors and 42 randomly selected psychology majors were observed to estimate the difference in how long it takes to graduate.

Find a 95% confidence interval for the difference.

We are 95% confident that the average time it takes to graduate is between 0.3 and 0.7 years longer for psychology majors than for engineer majors.


5.1, 0.4, 5.6, 0.5E E P Px s x s

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Hypothesis Test: Paired Samples

Does eating chocolate improve memory. 12 people were give a memory test before and after eating chocolate. The data for the number of words recalled out of 50 are shown below. Assume Normality.

1. Hypothesize H0: mdiff = 0, Ha: mdiff ≠ 0


Before 24 16 33 9 42 38 27 30 41

After 26 20 29 11 42 39 25 34 44

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Does eating chocolate improve memory. 12 people were give a memory test before and after eating chocolate. The data for the number of words recalled out of 50 are shown below. Assume Normality.2. Prepare

a = 0.05, T-Statistic, large sample

3. Compute to Compare Stat → T Statistics → Paired


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Does eating chocolate improve memory. 12 people were give a memory test before and after eating chocolate. The data for the number of words recalled out of 50 are shown below. Assume Normality.

4. Interpret P-value = 0.13 > 0.05 = a Fail to Reject H0

Conclusion: There is insufficient evidence to make a conclusion about the mean number of words increasing after eating chocolate.


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Hypothesis Test: Independent Samples

Do batteries last longer in colder climates than in warmer ones? The table shows some randomly selected battery lives in months.

1. Hypothesize H0: mF = mM

Ha: mF < mM


Florida 19 22 25 21 18 19 27 25 28 15

Montreal 37 49 22 26 47 41 38 37

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Preparea = 0.05Independent Samples, Assume Normal Distributions

Do batteries last longer in colder climates than in warmer ones?


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3. Compute to Compare Stat → T Statistics → Two sample → with data



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4. Interpret P-value = 0.0009 < 0.05 = a Reject H0

Accept Ha

Conclusion: There is statistically significance evidence to support the claim that on average batteries last longer in Montreal than in Florida.



Florida 19 22 25 21 18 19 27 25 28 15

Montreal 37 49 22 26 47 41 38 37


9.5

Overview of Analyzing Means

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General Formulas

Hypothesis Test Statistic

Confidence Interval


estimated value null hypothesis valueTest statistic

SE

: estimated value multiplier ESTCI SE

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Finding the p-value Given the Test Statistic Left Tailed Hypothesis:

Find the probability that a value is less than the test statistic .

Right Tailed Hypothesis: Find the probability that a value is greater than

the test statistic . Two Tailed Hypothesis:

Make the test statistic negative. Then find the probability that a value is less than the test statistic. Finally multiply by 2.


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Comparing CI and Hypothesis Tests

It can be concluded at the 5% level that the value is not the mean, proportion, or difference if a value falls outside the 95% confidence interval the p-value is less than 0.05

A 95% (90%, 99%) confidence interval is equivalent to a two-tailed test with a = 0.05 (0.1, 0.01) when it comes to rejecting or failing to reject H0.


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Hypothesis Tests and CI Example

Suppose that a hypothesis test: H0: m = 80 Ha: m ≠ 80was done for the average height of male college basketball players. If p-value = 0.02 can the 95% confidence interval contain 80? No. Since the p-value < 0.05, H0 is rejected. 80

cannot be in the confidence interval.


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Hypothesis Test or Confidence Interval: Which Should be Used?

For one-tailed testing: hypothesis test For two tailed testing: either can be used Confidence Intervals give more than

hypothesis tests. CI gives a plausible range for the population

value. The hypothesis test addresses the question of

whether H0 is false



Chapter 9

Case Study

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Epilepsy, Drugs, and Giving Birth

Four drugs are taken for epilepsy: carbamazepine, lamotrigine, phenytoin, and valproate.

Three years after pregnant mothers took the medicine, their children were given a IQ test.

The New England Journal of Medicine reported that taking valproate increased the risk of impaired cognitive development.


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95% Confidence Intervals

These give us a visual comparison. The valporate CI does not overlap with the

lamotrigine CI. For better comparisons, use confidence

intervals for the difference between means.


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Confidence Intervals for Differences

None contain 0. A hypothesis test for a difference between the means will reject H0.

There is statistically significant evidence to conclude that the mean IQ for children born to mothers taking valproate is different than for any of the other drugs.



Chapter 9

Guided Exercise 1

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Is the Mean Body Temperature really 98.6?

A random sample of 10 independent healthy people showed body temperatures (in degrees Fahrenheit) as follows: 98.5, 98.2, 99.0, 96.3, 98.3,

98.7, 97.2, 99.1, 98.7, 97.2 Use a = 0.05.

1. Hypothesize H0: m = 98.6

Ha: m ≠ 98.6


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2. Prepare

Not far from normal. Sample collected randomly. Use the t-statistic.


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3. Compute to Compare

t ≈ -1.65 p-value ≈ 0.13 p-value ≈ 0.13 > 0.05 = a


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4. Interpret

A random sample of 10 independent healthy people showed body temperatures (in degrees Fahrenheit) as follows: 98.5, 98.2, 99.0, 96.3, 98.3,

98.7, 97.2, 99.1, 98.7, 97.2 p-value = 0.13 > 0.05 = a We cannot reject 98.6 as the population mean

body temperature from these data at the 0.05 level.Copyright © 2013 Pearson Education, Inc.. All rights reserved.


Chapter 9

Guided Exercise 2

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A two-sample t-test for the number of televisions owned in households of random samples of students at two different community colleges. Assume independence. One of the schools is in a wealthy community (MC), and the other (OC) is in a less wealthy community.


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1. Hypothesize

Let moc be the population mean number of televisions owned by families of students in the less wealthy community (OC), and let mmc be the population mean number of televisions owned by families of students at in the wealthy community (MC).

H0: moc = mm

Ha: moc ≠ mm


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2. Prepare

Choose an appropriate t-test. Because the sample sizes are 30, the Normality condition of the t-test is satisfied. State the other conditions, indicate whether they hold, and state the significance level that will be used.

Use a t-test with two independent samples. The households were chosen randomly and

independently. The population of all households of each type

is more than 10 times the sample sizes.Copyright © 2013 Pearson Education, Inc.. All rights reserved.

1 - 65


t = 0.95 p-value = 0.345


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4. Interpret

Since the p-value = 0.345 is very large, we fail to reject H0.

At the 5% significance level, we cannot reject the hypothesis that the mean number of televisions of all students in the wealthier community is the same as the mean number of televisions of all students in the less wealthy community.



Chapter 9

Guided Exercise 3

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Pulse Before and After Fright

Test the hypothesis that the mean of college women’s pulse rates is higher after a fright, using a = 0.05.

1. Hypothesize H0: mbefore = mafter

Ha: mbefore > mafter


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2. Prepare

Choose a test: Should it be a paired t-test or a two-sample t-test? Why? Assume that the sample was random and that the distribution of differences is sufficiently Normal. Mention the level of significance. Paired t-test since before and after. Level of Significance: a = 0.05.


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t ≈ 4.9 p-value = 0.002 0.002 < 0.05


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4. Interpret

Reject or do not reject H0. Then write a sentence that includes “significant” or “significantly” in it. Report the sample mean pulse rate before the scream and the sample mean pulse rate after the scream. Reject H0. There is statistically significant evidence to

support the claim that mean blood pressure is higher after a fright.

mbefore ≈ 74.8

mafter ≈ 83.7


Copyright © 2013 Pearson Education, Inc. All rights reserved Chapter 9 Inferring Population Means.

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