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

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

2 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.

3 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.

4 Copyright © 2013 Pearson Education, Inc. All rights reserved 9.1 Sample Means of Random Samples

5 1 - 5 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.

6 1 - 6 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. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

7 1 - 7 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. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

8 1 - 8 Simulating Many Sample Means  As the sample size increases  Better Precision  Accuracy Does Not Change Copyright © 2013 Pearson Education, Inc.. All rights reserved.

9 1 - 9 Standard Error  The Standard Error is the standard deviation of the sampling distribution.   Copyright © 2013 Pearson Education, Inc.. All rights reserved.

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

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

12 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 Copyright © 2013 Pearson Education, Inc.. All rights reserved.  n = 1  n = 4  n = 16  n = 100 → $12,000 → $6,000 → $3,000 → $1,200

13 Copyright © 2013 Pearson Education, Inc. All rights reserved 9.2 The Central Limit Theorem for Sample Means

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

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

16 Central Limit Theorem For Means  Central Limit Theorem: If the conditions are met and the population has mean  and standard deviation , then the sampling distribution will be approximately normal. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

17 Visualizing the Central Limit Theorem  Population Distribution Skewed Right  Sampling Distribution Approximately Normal Copyright © 2013 Pearson Education, Inc.. All rights reserved.

18 Applying the Central Limit Theorem  The distribution of women’s pulse rates is skewed right with  = 74 bpm,  = 13 bpm.  If 30 women are selected, find   Copyright © 2013 Pearson Education, Inc.. All rights reserved.

19 Applying the Central Limit Theorem  The distribution of women’s pulse rates is skewed right with  = 74 bpm,  = 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. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

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

21 The t-Distribution  If  is unknown, we cannot find the z-score.  Use the sample standard deviation s instead.  is an estimate for the standard error Copyright © 2013 Pearson Education, Inc.. All rights reserved.

22 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.

23 Copyright © 2013 Pearson Education, Inc. All rights reserved 9.3 Answering Questions about the Mean of a Population

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

25 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: x 0.30 ≈ 9.5  (8.5,9.5) Copyright © 2013 Pearson Education, Inc.. All rights reserved.

26 CI Interpretations: (8.5,9.5)  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.  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. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

27 CI Interpretations: (8.5,9.5)  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.  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. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

28 Hypothesis Test for a Population Mean  The same four steps apply for a hypothesis test for a population mean: 1. Hypothesize: State H 0 and H a. 2. Prepare: Choose , 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 . 4. Interpret: Reject or fail to Reject H 0 ? Write down the conclusion in the context of the study. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

29 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  H 0 :  = 40, H a :  ≠ Prepare  Choose  = 0.05, Use t-statistic: random and large sample Copyright © 2013 Pearson Education, Inc.. All rights reserved.

30 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 <  = 0.05  Reject H 0. Accept H a. There is statistically significant evidence to conclude that your Focus does not get 40 mpg on average. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

31 Copyright © 2013 Pearson Education, Inc. All rights reserved 9.4 Comparing Two Population Means

32 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.

33 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. Copyright © 2013 Pearson Education, Inc.. All rights reserved. → Ind → Dep → Ind

34 Independent Samples Standard Error and Margin of Error   Degrees of Freedom is approximately the smaller of n 1 – 1 and n 2 – 1.  Use a computer or calculator for better accuracy. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

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

36 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.

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

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

39 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  H 0 :  diff = 0, H a :  diff ≠ 0 Copyright © 2013 Pearson Education, Inc.. All rights reserved. Before After

40 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   = 0.05, T-Statistic, large sample 3. Compute to Compare  Stat → T Statistics → Paired Copyright © 2013 Pearson Education, Inc.. All rights reserved.

41 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 =   Fail to Reject H 0  Conclusion: There is insufficient evidence to make a conclusion about the mean number of words increasing after eating chocolate. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

42 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  H 0 :  F =  M  Ha:  F <  M Copyright © 2013 Pearson Education, Inc.. All rights reserved. Florida Montreal

43  Prepare   = 0.05  Independent Samples,  Assume Normal Distributions Do batteries last longer in colder climates than in warmer ones? Copyright © 2013 Pearson Education, Inc.. All rights reserved.

44 Compute to Compare  Stat → T Statistics → Two sample → with data Do batteries last longer in colder climates than in warmer ones? Copyright © 2013 Pearson Education, Inc.. All rights reserved.

45 Interpret  P-value = < 0.05 =   Reject H 0  Accept H a  Conclusion: There is statistically significance evidence to support the claim that on average batteries last longer in Montreal than in Florida. Do batteries last longer in colder climates than in warmer ones? Copyright © 2013 Pearson Education, Inc.. All rights reserved. Florida Montreal

46 Copyright © 2013 Pearson Education, Inc. All rights reserved 9.5 Overview of Analyzing Means

47 General Formulas  Hypothesis Test Statistic  Confidence Interval Copyright © 2013 Pearson Education, Inc.. All rights reserved.

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

49 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  = 0.05 (0.1, 0.01) when it comes to rejecting or failing to reject H 0. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

50 Hypothesis Tests and CI Example  Suppose that a hypothesis test: H 0 :  = 80 H a :  ≠ 80 was 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, H 0 is rejected. 80 cannot be in the confidence interval. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

51 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 H 0 is false Copyright © 2013 Pearson Education, Inc.. All rights reserved.

52 Copyright © 2013 Pearson Education, Inc. All rights reserved Chapter 9 Case Study

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

54 % 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. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

55 Confidence Intervals for Differences  None contain 0. A hypothesis test for a difference between the means will reject H 0.  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. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

56 Copyright © 2013 Pearson Education, Inc. All rights reserved Chapter 9 Guided Exercise 1

57 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  = Hypothesize  H 0 :  = 98.6  H a :  ≠ 98.6 Copyright © 2013 Pearson Education, Inc.. All rights reserved.

58 Prepare  Not far from normal.  Sample collected randomly.  Use the t-statistic. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

59 Compute to Compare  t ≈  p-value ≈ 0.13  p-value ≈ 0.13 > 0.05 =  Copyright © 2013 Pearson Education, Inc.. All rights reserved.

60 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 =   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.

61 Copyright © 2013 Pearson Education, Inc. All rights reserved Chapter 9 Guided Exercise 2

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

63 Hypothesize  Let  oc be the population mean number of televisions owned by families of students in the less wealthy community (OC), and let  mc be the population mean number of televisions owned by families of students at in the wealthy community (MC).  H 0 :  oc =  m  H a :  oc ≠  m Copyright © 2013 Pearson Education, Inc.. All rights reserved.

64 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.

65 Compute to Compare  t = 0.95  p-value = Copyright © 2013 Pearson Education, Inc.. All rights reserved.

66 Interpret  Since the p-value = is very large, we fail to reject H 0.  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. Copyright © 2013 Pearson Education, Inc.. All rights reserved.

67 Copyright © 2013 Pearson Education, Inc. All rights reserved Chapter 9 Guided Exercise 3

68 Pulse Before and After Fright  Test the hypothesis that the mean of college women’s pulse rates is higher after a fright, using  =  1. Hypothesize  H 0 :  before =  after  H a :  before >  after Copyright © 2013 Pearson Education, Inc.. All rights reserved.

69 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:  = Copyright © 2013 Pearson Education, Inc.. All rights reserved.

70 Compute to Compare  t ≈ 4.9  p-value =  < 0.05 Copyright © 2013 Pearson Education, Inc.. All rights reserved.

71 Interpret  Reject or do not reject H 0. 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 H 0. There is statistically significant evidence to support the claim that mean blood pressure is higher after a fright.   before ≈ 74.8   after ≈ 83.7 Copyright © 2013 Pearson Education, Inc.. All rights reserved.


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