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Exam3_Practice - STAT 350 – Fall 2008 Final Exam Your...

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Unformatted text preview: STAT 350 – Fall 2008 Final Exam Your Name: ____________________________________ Your Seat: __________ Section Time (circle): 10:30 12:30 1:30 Note: • You are responsible for upholding the Honor Code of Purdue University. This includes protecting your work from other students. • Show your work on all questions. Unsupported work will not receive full credit. Credit will not be given for dumb luck. You do not need to show work for multiple choice questions. • Decimal answers should be exact or to at least four significant digits. • Unless otherwise stated, assume the significance level for any hypothesis test is 0.05. • Standard Normal (Z) and values/probabilities must be taken from the tables provided. Probabilities, p-values, critical values, etc., for χ2, t, and F distributions must also be taken from the tables provided, unless this information is available on SAS output provided, in which case the values from SAS should be used. • You are allowed the following aids: a one-page (8.5×11 inch) cheat sheet, a scientific calculator, and pencils. • Turn off and put away your cell phone before the exam begins! Question Points Possible Points Missed 1 51 2 10 3 23 4 16 Total 100 Final Score: _______________ / 100 1. Jason is a sociology major. For his senior thesis, Jason randomly selected a number of residents from his hometown to survey. He asked each subject a range of demographic questions. Among the questions he asked were: "How many years of schooling have you had?" and "What is your annual income?" Limiting his sample to just those 30 subjects who were no longer in school (that is, who had completed their schooling), the number of years of schooling ranged from 9 to 22 years (mean 15.4 years) and the annual incomes ranged from $28,984 to $61,267 (mean $44790). Using these 30 subjects, he conducted a regression analysis to explore whether the amount of schooling affects income. The SAS output from this analysis is given below. The REG Procedure Model: MODEL1 Dependent Variable: income Number of Observations Read Number of Observations Used 30 30 Analysis of Variance Source DF Sum of Squares Mean Square Model Error Corrected Total 1 28 29 297006095 1179094138 1476100233 297006095 42110505 Root MSE Dependent Mean Coeff Var 6489.26074 44790 14.48804 R-Square Adj R-Sq F Value Pr > F 7.05 0.0129 0.2012 0.1727 Parameter Estimates Variable Intercept years_school DF Parameter Estimate Standard Error t Value Pr > |t| 1 1 30207 946.97283 5617.60177 356.57432 5.38 2.66 <.0001 0.0129 a. (2 pts) What percent of the variation in incomes is explained by the linear relationship between income and schooling? b. (2 pts) What is the correlation between income and years of schooling? c. (5 pts) Based on the above analysis, what is the income you would expect for an individual from this town who has had 17 years of schooling? STAT 350 Final Exam Page 2 of 9 1 (continued) d. (2 pts) The 5th subject in this analysis had 17 years of schooling and has an annual income of $41,019. What is the value of the 5th residual? e. How much extra money should an individual in this town expect to earn for every additional year of school he or she has completed? (i) (2 pts) Give a point estimate. (ii) (5 pts) Give a 95% confidence interval f. (8 pts) Jason wants to determine if the relationship between years of schooling and annual income is "statistically significant"? (i) Give the value of the appropriate test statistic (ii) Give the degrees of freedom for that test statistic (iii) Give the p-value. (iv) Based on this, is the relationship "statistically significant"? Just answer "yes" or "no". g. (6 pts) Now assume that Jason wants to test the null hypothesis that years of schooling does not affect annual income (that is, average annual income does not change with an increase in the number of years of schooling) versus the alternative hypothesis that average annual income increases as the number of years of schooling increases. (i) Give the value of the appropriate test statistic (ii) Give the degrees of freedom for that test statistic (iii) Give the p-value. STAT 350 Final Exam Page 3 of 9 1 (continued) h. According to the Bureau of Labor Statistics, nationwide, average income increased $1750 for each additional year of schooling. Jason wants to compare his town to the national average. He will test the null hypothesis that the trend in his town is the same as the national average against the alternative that the trend in his town is different than the national average. (i) (4 pts) Give the value of the appropriate test statistic (ii) (2 pts) Give the degrees of freedom for that test statistic (iii) (2 pts) Give the p-value. i. (6 pts) Give a 95% confidence interval for the true mean income of all residents of this town (who have completed their schooling). j. (5 pts) Based solely on the preceding analysis, would it be appropriate for Jason to conclude that additional schooling causes increased income? Justify your answer. STAT 350 Final Exam Page 4 of 9 2. Short Answer a. (5 pts) Think back to the experiment done on mice we used in Lectures 18 and 19. Mice were randomly assigned to receive varying doses of alcohol and then timed running a maze. The correlation between the dose of alcohol and the time to run the maze was 0.91626. Would it be appropriate to conclude that increased doses of alcohol causes an increase in the time to run the maze? Justify your answer. b. (5 pts) Define "p-value". You shouldn't need more space than this! STAT 350 Final Exam Page 5 of 9 Histogram of var1 Nor mal 9 Mean 8.5 StD ev 2.038 N 40 8 7 6 Frequency 3. Background: In Lab #3 we looked at a number of different ways to assess whether a sample may have come from a Normal distribution. Probability or QQ-plots are one approach. We also looked at a hypothesis test (the AndersonDarling Test) provided by Minitab along with the probability plot. Another approach we used was to visually compare the sample histogram with a normal distribution curve with the μ = x and σ = s (for example, see Figure 1). Another approach is to quantitatively compare the sample histogram to the normal distribution curve. A chi-squared test can then be used to determine whether the observed frequencies in the various bins are close enough to the frequencies that would be expected if the sample did come from a normal distribution. 5 4 3 2 1 0 4 6 8 var1 10 12 Figure 1. Histogram with Normal Distribution curve from Lab #3, Problem #1. Note, this is not the data for the problem below) A sample of 35 observations was taken. You wish to assess whether these observations may have come from a Normal distribution. The sample had x = 100 and s = 10. Of the 35 observations, 7 were less than 90, 10 were between 90 and 100, 10 were between 100 and 110, and the remaining 8 observations were greater than 110 * (see table below). bin <90 90 to 100 100 to 110 >110 observed frequency 7 10 10 8 a. (8 pts) The null hypothesis is that the sample did come from a normal distribution with μ = 100 and σ = 10. This null hypothesis can be restated in terms of the true proportions of each category, the πi's. Give the values of the πi's (to 4 decimal places) Hint: π1 = P(X < 90), where X ~ Normal(μ = 100 and σ = 10). Another hint: π1 + π2 + π3 + π4 = 1. π1 = _________________ π2 = _________________ π3 = _________________ π4 = _________________ * Note: Ideally more bins would be used, but then the analysis would take a lot longer (this is me being nice). Also, the data were continuous, so there is no problem with observations falling on the cut-point for a bin (e.g., no observation was exactly 100). STAT 350 Final Exam Page 6 of 9 3 (continued) b. (4 pts) Give the expected counts (to 4 decimal places) for each bin (fill in the table below). bin <90 90 to 100 100 to 110 >110 Expected Counts c. (4 pts) Give the value of the chi-square statistic for this analysis. d. (2 pts) What is the critical value for this test (α = 0.05)? e. (5 pts) Based on your analysis above, what is your conclusion? Circle one of the options below. The sample did come from a normal distribution The sample did NOT come from a normal distribution STAT 350 Final Exam Page 7 of 9 4. Susan is majoring in wood science. For her senior thesis, she wanted to examine the durability of wood used for decking. Deterioration of wood in use is commonly caused by decay fungi, certain insects (including termites) as well as other organisms, and weathering. She wanted to examine various species of wood and various preservatives. She used 4 species of wood: Red Oak, Spruce, Eastern Red Cedar, and Redwood. She used 3 types of preservatives: creosote, pentachlorophenol (PCP), and ammoniacal copper arsenate (ACA). She used 2 boards for each species-preservative combination. The boards were dried and weighed then placed outdoors (all in the same experimental plot) for 10 months. At the end of 10 months, the boards were collected, dried, and re-weighed. For each board, the dry weight-loss (in grams) was recorded. The SAS results from the ANOVA are given below. Some information has been omitted and replaced with asterisks (*). The SAS System 1 The ANOVA Procedure Class Level Information Class Levels Values species 4 RedCedar RedOak Redwood Spruce preserv 3 ACA Creosote PCP Number of Observations Read Number of Observations Used 24 24 The SAS System 2 The ANOVA Procedure Dependent Variable: loss Source DF Sum of Squares Mean Square F Value Pr > F Model ** 8948.33833 ********* ***** ****** Error ** 3483.66000 ********* Corrected Total ** 12431.99833 R-Square Coeff Var Root MSE loss Mean 0.719783 37.85597 17.03834 45.00833 Source DF Anova SS Mean Square F Value Pr > F species preserv species*preserv ** ** ** 5272.615000 2371.890833 1303.832500 *********** *********** *********** ***** ***** ***** ****** ****** ****** STAT 350 Final Exam Page 8 of 9 4 (continued) a. Is there a statistically significant difference among the 3 preservatives? (i) (4 pts) Give the value of the appropriate test statistic. (ii) (2 pts) Give the degrees of freedom for this test statistic. (iii) (2 pts) Give the critical value for this test (α = 0.05). (iv) (4 pts) Based on your analysis above, what is your conclusion? Circle one of the options below. There is NO statistically significant difference in the mean weight loss among the 3 types of preservatives studied The mean weight loss for each preservative is significantly different from each of the other preservatives. At least one of the preservatives has a mean weight loss that is significantly different from the other preservatives. b. (4 pts) Assume that the p-value for "species" had been large (e.g., 0.80), but the p-value for "species*preserv" had been small (e.g., 0.001) ** . Would it then be reasonable to expect the same performance (loss) for all species of wood for a given preservative? Imagine you are a statistical consultant trying to explain the implication of these results to Susan. ** The p-values given for part (b) are not actually the p-values from the analysis above. But you are to assume they are for the sake of answering part (b) STAT 350 Final Exam Page 9 of 9 ...
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