Calculating Degrees of Freedom The confidence interval

Problem 1BSC Independent and Dependent Samples Which of the following involve independent samples? a. To test the effectiveness of the Atkins diet, 36 randomly selected subjects are weighed before the diet and six months after treatment with the diet. The two samples consist of the before/after weights. ________________ b. To determine whether smoking affects memory, 50 randomly selected smokers are given a test of word recall and 50 randomly selected nonsmokers are given the same test. Sample data consist of the scores from the two groups. ________________ c. IQ scores are obtained from a random sample of 75 wives and IQ scores are obtained from their husbands. ________________ d. Annual incomes are obtained from a random sample of 1200 residents of Alaska and from another random sample of 1200 residents of Hawaii. ________________ e. Scores from a standard test of mathematical reasoning are obtained from a random sample of statistics students and another random sample of sociology students.

Interpreting Confidence Intervals If the heights of men and women from Data Set 1 in Appendix B are used to construct a 95% confidence interval for the difference between the two population means, the result is \(\mathrm {11.61\ cm< \mu\ 1 - mu\ 2<17.32\ cm}\), where heights of men correspond to population 1 and heights of women correspond to population 2. Express the confidence interval with heights of women being population 1 and heights of men being population 2. ________________ Equation Transcription: Text Transcription: 11.61 cm< mu 1 - mu 2<17.32 cm

Interpreting Confidence Intervals What does the confidence interval in Exercise 2 suggest about the heights of men and women?

Problem 4BSC Hypothesis Tests and Confidence Intervals a. In general, if you conduct a hypothesis test using the methods of Part 1 of this section, will the P-value method, the critical value method, and the confidence interval method result in the same conclusion? ________________ b. Assume that you want to use a 0.05 significance level to test the claim that the mean height of men is greater than the mean height of women. What confidence level should be used if you want to test that claim using a confidence interval?

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Color and Creativity Repeat Example 2 with these changes: Use a 0.05 significance level, and test the claim that the two samples are from populations with the same mean.

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­-value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Color and Cognition In the Chapter Problem, it was noted that researchers conducted a study to investigate the effects of color on cognitive tasks. Words were displayed on a computer screen with background colors of red and blue. Results from scores on a test of word recall are given below. Use a 0.05 significance level to test the claim that the samples are from populations with the same mean. Does the background color appear to have an effect on word recall scores? If so, which color appears to be associated with higher word memory recall scores? Red Background: n = 35 , \(\mathrm {x\ ^-=15.89}\) , s = 5.90 Blue Background: n = 36 , \(\mathrm {x\ ^-=12.31}\) , s = 5.48 ________________ Equation Transcription: Text Transcription: n=35 x ^{-} =15.89 s=5.90 n=36 x ^{-} =12.31 s=5.48

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­-value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Magnet Treatment of Pain People spend around $5 billion annually for the purchase of magnets used to treat a wide variety of pains. Researchers conducted a study to determine whether magnets are effective in treating back pain. Pain was measured using the visual analog scale, and the results given below are among the results obtained in the study (based on data from “Bipolar Permanent Magnets for the Treatment of Chronic Lower Back Pain: A Pilot Study,” by Collacott, Zimmerman, White, and Rindone, Journal of the American Medical Association, Vol. 283, No. 10). Use a 0.05 significance level to test the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment (similar to a placebo). Does it appear that magnets are effective in treating back pain? Is it valid to argue that magnets might appear to be effective if the sample sizes are larger? Reduction in Pain Level After Magnet Treatment: n = 20 , \(\mathrm {x\ ^-=0.49}\) , s = 0.96 Reduction in Pain Level After Sham Treatment: n = 20 , \(\mathrm {x\ ^-=0.44}\) , s = 1.4 ________________ Equation Transcription: Text Transcription: n=20 x ^-=0.49 s=0.96 n=20 x ^-=0.44 s=1.4

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­-value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Do Men Talk Less Than Women? The accompanying table gives results from a study of the words spoken in a day by men and women, and the original data are in Data Set 17 in Appendix B (based on “Are Women Really More Talkative Than Men?” by Mehl, et al., Science, Vol. 317, No. 5834). Use a 0.01 significance level to test the claim that the mean number of words spoken in a day by men is less than that for women. ________________ Equation Transcription: Text Transcription: n 1=186 x -=15,668.5 s 1=8632.5 n 2=210 x - 2=16,215.0 s 2=7301.2

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­-value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Do Men Have a Higher Mean Body Temperature? If we use the body temperatures from 8 A.M. on Day 2 as listed in Data Set 3 in Appendix B, we get the statistics given in the accompanying table. Use these data with a 0.01 significance level to test the claim that men have a higher mean body temperature than women. ________________ Equation Transcription: Text Transcription: n 1=11 x ^{-} 1=97.69 ^o F s 1=0.89 ^o F n 2=59 x ^{-} 2=97.45 ^o F s 2=0.66 ^o F

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­-value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Do Men and Women Have the Same Mean Body Temperature? Consider the sample of body temperatures \(\mathrm {(\ ^\circ\ F\ )}\) listed in the last column of Data Set 3 in Appendix B. The summary statistics are given in the accompanying table. Use a 0.01 significance level to test the claim that men and women have different mean body temperatures. ________________ Equation Transcription: Text Transcription: ( ^o F ) n 1=15 x ^- 1=93.38 ^o F s 1 =0.45 ^o F n 2 =91 x ^- 2=98.17 ^o F s 2=0.65 ^o F

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­-value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Skull Measurements from Different Times Researchers measured skulls from different time periods in an attempt to determine whether interbreeding of cultures occurred. Results are given below (based on data from Ancient Races of the Thebaid, by Thomson and Randall­Maciver, Oxford University Press). Use a 0.01 significance level to test the claim that the mean maximal skull breadth in 4000 B.C. is less than the mean in A.D. 150. 4000 B.C. ( Maximal Skull Breadth ) : n = 30 , \(\mathrm {x\ ^{-}=131.37\ \ mm}\) , s = 5.13 mm A.D. 150 ( Maximal Skull Breadth ) : n = 30 , \(\mathrm {x\ ^{-}=136.17\ \ mm}\) , s= 5.35 mm ________________ Equation Transcription: Text Transcription: n=30 x ^{-}=131.37 mm s=5.13 mm A.D. 150 n=30 x ^{-}=136.17 mm s=5.35 mm

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­-value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Flight Arrival Delays Data Set 15 in Appendix B lists arrival delay times (min) for randomly selected flights from New York (JFK) to Los Angeles (LAX). Statistics for times are given below. Use a 0.05 significance level to test the claim that Flight 1 and Flight 3 have the same mean arrival delay time. Flight 1 n = 12 , \(\mathrm {x\ ^{-}=-20.5\ \ min}\) , s = 12.38401 min Flight 3 n = 12 , \(\mathrm {x\ ^{-}=-15.08333\ \ min}\) , s = 15.62317 min ________________ Equation Transcription: Text Transcription: 1 n = 12 x ^{-}=-20.5 min s=12.38401 min 3 n=12 x ^{-}=-15.08333 min s=15.62317 min

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P-­value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Proctored and Nonproctored Tests In a study of proctored and nonproctored tests in an online Intermediate Algebra course, researchers obtained the data for test results given below (based on “Analysis of Proctored versus Non­proctored Tests in Online Algebra Courses,” by Flesch and Ostler, MathAMATYC Educator, Vol. 2, No. 1). Use a 0.01 significance level to test the claim that students taking nonproctored tests get a higher mean than those taking proctored tests. Group 1 ( Proctored ) : n = 30 , \(\mathrm {x\ ^{-}=74.30}\) , s = 12.87 Group 2 ( Nonproctored ) : n = 32 , \(\mathrm {x\ ^{-}=88.62}\) , s = 22.09 ________________ Equation Transcription: Text Transcription: n=30 x ^{-}=74.30 s=12.87 n=32 x ^{-}=88.62 s=22.09

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­-value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Proctored and Nonproctored Tests In the same study described in the preceding exercise, the same groups of students took a nonproctored test; the results are given below. Use a 0.01 significance level to test the claim that the samples are from populations with the same mean. Group 1 ( Nonroctored ) : n = 30 , \(\mathrm {x\ ^{-}=70.29}\) , s = 22.09 Group 2 ( Nonproctored ) : n = 32 , \(\mathrm {x\ ^{-}=74.26}\) , s = 18.15 ________________ Equation Transcription: Text Transcription: n=30 x ^{-}=70.29 s=22.09 n=-32 x ^{-}=74.26 s=18.15

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P-­value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. BMI We know that the mean weight of men is greater than the mean weight of women, and the mean height of men is greater than the mean height of women. A person's body mass index (BMI) is computed by dividing weight (kg) by the square of height (m). Given below are the BMI statistics for random samples of males and females from Data Set 1 in Appendix B. Use a 0.05 significance level to test the claim that males and females have the same mean BMI. Male BMI n = 40 , \(\mathrm {x\ ^{-}=28.44075}\) , s = 7.394076 Female BMI n = 40 , \(\mathrm {x\ ^{-}=26.6005}\) , s = 5.359442 ________________ Equation Transcription: Text Transcription: n=40 x ^{-}=28.44075 s=7.394076 n=40 x ^{-}=26.6005 s=5.359442

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­-value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. IQ and Lead Exposure Data Set 5 in Appendix B lists full IQ scores for a random sample of subjects with low lead levels in their blood and another random sample of subjects with high lead levels in their blood. The statistics are summarized below. Use a 0.05 significance level to test the claim that the mean IQ score of people with low lead levels is higher than the mean IQ score of people with high lead levels. Low Lead Level: n = 78 , \(\mathrm {x\ ^{-}=92.88462}\) , s = 15.34451 High Lead Level: n = 21 , \(\mathrm {x\ ^{-}=86.90476}\) , s = 8.988352 ________________ Equation Transcription: Text Transcription: n = 78 x ^{-}= 92.88462 s = 15.34451 n = 21 x ^{-}= 86.90476 s = 8.988352

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. IQ and Lead Exposure Repeat Exercise 16 after replacing the low lead level group with the following full IQ scores from the medium lead level group.

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P-­value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Heights of Supermodels Listed below are the heights (inches) for the simple random sample of supermodels Lima, Bundchen, Ambrosio, Ebanks, Iman, Rubik, Kurkova, Kerr, Kroes, and Swanepoel. Data Set 1 in Appendix B includes the heights of a simple random sample of 40 women from the general population, and here are the statistics for those heights: n = 40 , \(\mathrm {x\ ^{-}=63.7815\ in}\)., and s = 2.59665 in. Use a 0.01 significance level to test the claim that the supermodels have heights with a mean that is greater than the mean height of women in the general population. ________________ Equation Transcription: Text Transcription: n=40 x -=63.7815 in s=2.59665 in

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Longevity Listed below are the numbers of years that popes and British monarchs (since 1690) lived after their election or coronation (based on data from Computer­Interactive Data Analysis, by Lunn and McNeil, John Wiley & Sons). Treat the values as simple random samples from a larger population. Use a 0.01 significance level to test the claim that the mean longevity for popes is less than the mean for British monarchs after coronation.

In Exercises 5–20, assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Do the following: a. Test the given claim using the P­value method or critical value method. b. Construct a confidence interval suitable for testing the given claim. Radiation in Baby Teeth Listed below are amounts of strontium­90 (in millibecquerels, or mBq, per gram of calcium) in a simple random sample of baby teeth obtained from Pennsylvania residents and New York residents born after 1979 (based on data from “An Unexpected Rise in Strontium­90 in U.S. Deciduous Teeth in the 1990s,” by Mangano, et al., Science of the Total Environment, Vol. 317). Use a 0.05 significance level to test the claim that the mean amount of strontium­90 from Pennsylvania residents is greater than the mean amount from New York residents.

Large Data Sets. In Exercises 21–24, use the indicated Data Sets from Appendix B. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Weights of Quarters Vending machines reject coins based on weight. Refer to Data Set 21 in Appendix B and use a 0.05 significance level to test the claim that the mean weight of pre­1964 quarters is equal to the mean weight of post­1964 quarters. Given the relatively small sample sizes from the large populations of millions of quarters, can we really conclude that the mean weights are different?

Large Data Sets. In Exercises 21–24, use the indicated Data Sets from Appendix B. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Baseline Characteristics Reports of results from clinical trials often include statistics about “baseline characteristics,” so we can see that different groups have the same basic characteristics. Refer to Data Set 1 in Appendix B and construct a 95% confidence interval estimate of the difference between the mean age of men and the mean age of women. Based on the result, does it appear that the sample of men and the sample of women are from populations with the same mean?

Large Data Sets. In Exercises 21–24, use the indicated Data Sets from Appendix B. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Weights of Pepsi Refer to Data Set 19 in Appendix B and construct a 95% confidence interval estimate of the difference between the mean weight of the cola in cans of regular Pepsi and the mean weight of cola in cans of Diet Pepsi. Does there appear to be a difference between those two means? If there is a difference in the mean weights, identify the most likely explanation for that difference.

Large Data Sets. In Exercises 21–24, use the indicated Data Sets from Appendix B. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal Weights of Coke Refer to Data Set 19 in Appendix B and use a 0.05 significance level to test the claim that because they contain the same amount of cola, the mean weight of cola in cans of regular Coke is the same as the mean weight of cola in cans of Diet Coke. If there is a difference in the mean weights, identify the most likely explanation for that difference.

Pooling. In Exercises 25 and 26, assume that the two samples are independent simple random samples selected from normally distributed populations. Also assume that the population standard deviations are equal \((\ \sigma\ 1=\sigma\ 2\ )\) , so that the standard error of the differences between means is obtained by pooling the sample variances as described in Part 2 of this section. Do Men Have a Higher Mean Body Temperature? Repeat Exercise 9 with the additional assumption that \(\sigma\ 1=\sigma\ 2\) . How are the results affected by this additional assumption? ________________ Equation Transcription: Text Transcription: ( sigma 1= 2 ) sigma 1= 2

Pooling. In Exercises 25 and 26, assume that the two samples are independent simple random samples selected from normally distributed populations. Also assume that the population standard deviations are equal \((\ \sigma\ 1=\sigma\ 2\ )\) , so that the standard error of the differences between means is obtained by pooling the sample variances as described in Part 2 of this section. Do Men Talk Less Than Women? Repeat Exercise 8 with the additional assumption that \(\sigma\ 1=\sigma\ 2\) . How are the results affected by this additional assumption? ________________ Equation Transcription: Text Transcription: ( sigma 1= 2 ) sigma 1= 2

No Variation in a Sample An experiment was conducted to test the effects of alcohol. Researchers measured the breath alcohol levels for a treatment group of people who drank ethanol and another group given a placebo. The results are given below. Use a 0.05 significance level to test the claim that the two sample groups come from populations with the same mean. The given results are based on data from “Effects of Alcohol Intoxication on Risk Taking, Strategy, and Error Rate in Visuomotor Performance,” by Streufert, et al., Journal of Applied Psychology, Vol. 77, No. 4. Treatment Group: n 1 = 22 , \(\mathrm {x\ ^{-}1=0.049}\) , s 1 = 0.015 Placebo Group: n 2 = 22 , \(\mathrm {x\ ^{-} 2=0.000}\) , s 2 = 0.000 ________________ Equation Transcription: Text Transcription: n 1=22 x ^{-}1=0.049 s 1=0.015 n 2=22 x ^{-} 2=0.000 s 2=0.000

Calculating Degrees of Freedom The confidence interval given in Exercise 2 is based on df = 39 , which is the “smaller of n 1 ? 1 and n 2 ? 1. ” Use Formula 9­1 to find the number of degrees of freedom. Using the number of degrees of freedom from Formula 9­1 results in this confidence interval: \(\mathrm {11.65\ \ cm\ <\mu\ 1\ - \mu\ 2<17.28\ \ cm}\) . In what sense is “ df = smaller of n 1 ? 1 and n 2 ? 1 ” a more conservative estimate of the number of degrees of freedom than the estimate obtained with Formula 9­1? ________________ Equation Transcription: Text Transcription: df=39 n 1-1 n 2-1 11.65 cm < mu 1- mu 2<17.28 cm df=smaller n 1-1 n 2-1

One Standard Deviation Known We sometimes know the value of one population standard deviation from an extensive history of data, but a new procedure or treatment results in sample values with an unknown standard deviation. If \(\sigma\ 1\) is known but \(\sigma\ 2\) is unknown, use the procedures in Part 1 of this section with these changes: Replace s 1 with the known value of \(\sigma\ 1\) and find the number of degrees of freedom using the expression below. (See “The Two­Sample t Test with One Variance Unknown,” by Maity and Sherman, The American Statistician, Vol. 60, No. 2.) Repeat Exercise 13 by assuming that \(\sigma=15\) for proctored tests. \(\mathrm {df=(\ \sigma\ 1\ 2\ n\ 1 + s\ 2\ 2\ n\ 2\ )\ 2\ (\ s\ 2\ 2\ /\ n\ 2\ )\ 2\ n\ 2 - 1}\) ________________ Equation Transcription: Text Transcription: sigma 1 sigma 2 sigma 1 sigma=15 df=( sigma 1 2 n 1 + s 2 2 n 2 ) 2 ( s 2 2 / n 2 ) 2 n 2 - 1