MidtermI_-2_25_2008_sol

MidtermI_-2_25_2008_sol - Midterm Exam #1, 2008, Stat 102...

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1 Midterm Exam #1, 2008, Stat 102 – Solutions Sections: 001 = M&W@1:30; 002 = T&Th@10:30; 003 = T&Th@1:30 Instructions : This is a semi-closed book exam. Up to two pages of notes are allowed. (These two pages may be two-sided if you wish.) When performing hypothesis tests, clearly state the null and alternative hypotheses and show the critical value and/or P-value (from the table) where appropriate. Normal tables, t tables and F tables are provided, as needed. Write your answers on the test and always show your work. Partial credit will be given. Time: 90 minutes . Your Name ____________________________ Section #: 001 (Prof. Brown) _______ 002 (Prof. Zhao) _______ 003 (Prof. Zhao) _______ Sections: 001 = M&W@1:30; 002 = T&Th@10:30; 003 = T&Th@3:00 Point Allocations are indicated for each numbered problem (out of 100 total points).
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2 1. [20 Points] It is often claimed that blood pressure increases with age. Here is some data about Systolic Blood Pressure for a sample of females at age 48 and at age 58. Here are the side- by-side boxplots and summary statistics for the 48 year olds and for the 58 year olds. 100 150 200 250 SBP 48 58 AGE Age Number Mean Std Dev Std Err Mean 48 41 142.85 20.48 3.198 58 37 157.14 33.38 5.488 Here is some JMP output for tests of equality of blood pressure among the two age groups: Assuming equal variances Difference 14.282 t Ratio 2.302 Std Err Dif 6.204 DF 76 Not Assuming equal variances Difference 14.282 t Ratio 2.248 Std Err Dif 6.352 DF 58.5 (a) Conduct a suitable test of the null hypothesis that the two groups have equal mean blood pressures. Give the (approximate) P-value of the test. Can you conclude at α = 0.05 that the two groups have different mean blood pressures? Can you conclude this to be so at level α = 0.01? SOLUTION: Since the larger of the larger ratio of 2 2 2 2 12 / 33.38 / 20.48 2.67 3 ss , we may take the equal variance assumption. So the t ratio is 2.3 with df=76. This results in a p-value of 2*P(Z>2.3)=2*.0107=.02. We will reject the null hypothesis at α = 0.05 but not at α = 0.01.
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3 A more careful statistician noticed that the same women were measured at age 48 and again at age 58 (except that there were 4 women who were only measured at age 48). This statistician calculated the differences between the blood pressures of each woman at age 58 and age 48. He thus calculated 37 differences – one for each of these women. Here are summary statistics for these 37 differences: Difference in SBP Mean 16.919 Std Dev 28.773 Std Err Mean 4.730 N 37 (b) Use this information to construct a better test of the null hypothesis that blood pressure (among women) does not change on average from age 48 to age 58? Perform the test and give its (approximate) P-value. Can you conclude at α = 0.05 that mean blood pressure for women changes during this age range? Can you conclude this to be so at level α = 0.01? SOLUTION: The test statistic is 16.919 4.730 3.577 . We have a t distribution with 36 degrees of freedom. Using the table with 40 degrees of freedom, we find that 3.577 is beyond the .005 upper tail cutoff, so p<.01, and we reject the null at =.05 and =.01.
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4 2. [40 points] The following is based on a data set that contains the mileage and asking price for a random selection of 162 used cars. (The data is from a California website about 2 years ago. Some of the data was missing for some of the cars, and so the following analysis
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This note was uploaded on 04/04/2012 for the course STAT 102 taught by Professor Shaman during the Spring '08 term at UPenn.

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MidtermI_-2_25_2008_sol - Midterm Exam #1, 2008, Stat 102...

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