Stats 350 Fall 2009 Final Exam Solutions 1. Fashion Trends: Bella wants to study the fashion tendencies at her new school in Folks, WA where the weather is usually somewhat cool. During lunch one day she conducted an observational study to learn about what some of the ?hip? fashion trends were for females. Below is a summary of what she observed: Type of Footwear # wearing Fur Boots # Wearing Sneakers # wearing jeans 34 86 # wearing skirts X 4 a. If a girl wearing sneakers is randomly selected, what is the probability that she is wearing them with a skirt?  4/90 = 0.0444 Final Answer = ___0.0444 ___ b. Bella read in Vogue that wearing fur boots and wearing a skirt were mutually exclusive. Assuming Vogue is correct, what is the value for the missing cell value X?  If mutually exclusive (or disjoint) there the intersection is empty, so the value is 0 Final Answer = ___ 0___ c. Based on the results above and the answer in part (b), the events wearing fur boots and wearing a skirt are:  Circle one: dependent independent 2. Nausea or Not? ? An article in Ergonomics described a study designed to assess if the incidence of nausea is the same across the three seat locations in a bus (front, middle, or rear). The following categorical data were available from this study. Front Middle Rear Total Nausea 58 166 193 417 No nausea 870 1163 806 2839 Total 928 1329 999 3256 a. Fill in the blank:  In this study the variable seat location plays the role of the ___explanatory___ variable. b. Give the name of the test for assessing if the incidence of nausea is the same across the three seat location populations.  Name of the test: ___Chi?square test of HOMOGENEITY____ c. If there is no difference in the incidence of nausea across the three seat location populations, how many of the middle of the bus passengers would be expected to experience nausea? (show your work)  (417*1329)/3256 = 170.21 Final Answer: ___170.21_________ d. The observed counts were entered into the statistical package R and the test output is provided at the right. Circle the correct words to complete the conclusion based on the results of this study.  Thus it appears that the incidence of nausea is is not same for the 3 bus seat location populations. Pearson's Chi?squared test data: motion X?squared = 73.7891, df = 2, p?value < 2.2e?16 Stats 350 F09 Final Exam Solutions Page 1 of 10 Stats 350 Fall 2009 Final Exam Solutions 3. Responding favorably to a Treatment ? A medical researcher would like to test the hypothesis that a majority of patients who receive a new treatment will respond favorably, that is, test H 0 : p = 0.5 versus H a : p > 0.5, where p is the population proportion of all patients favorably responding to the new treatment. If there is not enough evidence for a majority, there will be no further funding of this treatment. There are 50 patients available for the study. a. The summary report stated that 56% of the treated patients responded successfully to the new treatment. How many of the patients responded favorably?  56% of 50 = 28 Final Answer = ____ 28____ b. The numerical value of 56% is a statistic which estimates a parameter. Circle one: True False  c. Using the results stated in part (a), perform the hypothesis test at a 5% level. Give the observed test statistic valu v e d e w all work. e, its respective p? alu , an d cision. Sho nullnull null.nullnullnullnull.nullnull null null.nullnullnullnullnullnull.nullnullnull nullnull null null.nullnull null.nullnull null null.nullnullnullnull so p?value = P(Z ? 0.85) = 0.1977.  Observed test statistic = __0.8545____ p?value = ___0.1977________ At a 5% level the statistical decision would be: (circle one) Reject H 0 Fail to reject H 0 d. Suppose this decision in part (c) is incorrect. What is a consequence of such a mistake?  Circle as many as apply: ? Patients may receive a treatment that doesn?t work. ? Patients may not receive a beneficial treatment. ? The company that sponsored the development of this treatment may not financially benefit. e. Determine if each of the following statements is true or false. [2 points each] i. By choosing a 5% level, the researcher accepts a 5% risk of incorrectly concluding a majority would respond favorably. True False ii. Power of a test is the probability that you correctly conclude a majority would respond favorably. True False Stats 350 F09 Final Exam Solutions Page 2 of 10 Stats 350 Fall 2009 Final Exam Solutions 4. Treating Migraines: A drug company is interested in determining if a new medication is effective in relieving pain associated with migraine headaches. A random sample of 20 migraine sufferers is obtained. Each patient rates his/her pain level (on a quantitative scale between 0 and 10) both before and after taking the medication. A rating of 10 represents the highest amount of pain possible and a rating of 0 represents no pain. For each person, the difference is defined to be pain level before taking the medication minus pain level after taking the medication. The company wants to determine if pain levels improve after taking the medication, on average. a. Clearly define the parameter of interest and then state the hypotheses to be tested.  Let the parameter __? d ___ = population mean difference in pain level (before less after) for the population of all such patients taking the new medication. The hypotheses to be tested are H 0 :__ ? d = 0 _____ versus H a :__ ? d > 0 _____________ The differences were entered into SPSS and the results for one sample t?test on the differences were obtained. One-Sample Statistics N Mean Std. Deviation Std. Error Mean b. Give the symbol and value for the test statistic to test the hypotheses in part (a).  t = (1.185)/(0.47191) = 2.511 ___ t ___ = ____2.511____ c. Provide the p?value for testing the hypotheses stated in part (a), circle your decision at the 5% level, and state your conclusion in the context of the problem.  p?value = _0.021/2 = 0.0105_ Decision? (circle one): Fail to Reject H 0 Reject H 0 Conclusion: There is sufficient evidence to say pain levels improve on average after taking the new medication. Or There is sufficient evidence to say the population mean Difference in headache pain is greater than 0. Or There is sufficient evidence to say on average, pain relief is significantly reduced using the new medication. d. The plot at the right was also produced. Clearly state the assumption being assessed through this plot.  That the population of differences is normally distributed. One-Sample Test Test Value = 0 t df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper Difference 2.511 19 .021 1.18500 .1973 2.1727 Difference 20 1.1850 2.11044 .47191 Stats 350 F09 Final Exam Solutions Page 3 of 10 Stats 350 Fall 2009 Final Exam Solutions 5. Does a Car?s Color Influence the Chance of Being Stolen? ? According to the American Automobile Association, 15% of all cars are white, 15% are blue, 35% are red, 30% are black, and the remaining 5% are some other color. The Associated Press reported the following color results for a random sample of 830 stolen cars: 1 = White 2 = Blue 3 = Red 4 = Black 5 = Other TOTAL # of stolen cars 140 100 270 230 90 830 The hypotheses to be tested using the above results are: H 0 : p 1 = 0.15, p 2 = 0.15, p 3 = 0.35, p 4 = 0.30, p 5 = 0.05 H a : at least one of the p i is not as listed in H 0 where p 1 is the proportion of all stolen cars that are white, p 2 is the proportion of all stolen cars that are blue, and so on. a. Give the name of the statistical test for testing the above hypotheses.  Name of the test: _______ Chi?square test of GOODNESS of FIT _______ b. If the null hypothesis is true, how many cars in the random sample would be expected to fall in the ?other? category? Show your work.  830 (0.05) = 41.5 Final Answer: ______41.5_______ c. The test statistic value is computed to be 66.33. Determine the contribution to this 66.33 that came from the ?other? category. Show all work.  (90 ? 41.5) 2 = 56.68 41.5 Final Answer: _____56.68______ d. What distribution would be used to find the corresponding p?value?  Final Answer: ___ ? 2 (4) distribution______ e. The researcher conducting the test noted that the observed test statistic value of 66.33 is highly significant as it is more than 2 standard deviations above the expected value for such a test statistic. What number represents being 2 standard deviations above the expected value in this case?  Show your work. 4 + 2(sqrt(2*4)) = 4 + 2(sqrt(8)) = 4 + 2(2.83) = 9.66 Final Answer: ______9.66_______ Stats 350 F09 Final Exam Solutions Page 4 of 10 Stats 350 Fall 2009 Final Exam Solutions 6. Tires: A consumer advocacy group wants to assess the difference between how long Michelin and Goodyear tires last, on average. A random sample of 30 Michelin Tires was obtained, and an independent, random sample of 35 Goodyear tires was obtained. The number of miles that each tire lasted was recorded for each tire. The following table summarizes the data that was collected (group 1 = Michelin, group 2 = Goodyear). Group 1 Group 2 Sample Mean 3120 3050 Sample Size 30 35 Sample Standard Deviation 200 195 a. The pooled confidence interval assumes that the two populations have equal variances. Citing appropriate evidence, comment on the validity of this assumption.  The two sample standard deviations of 200 and 195 are similar. b. An estimate of the common population standard deviation using the above results is 197.3. Provide a 95% pooled confidence interval estimate for the difference between the two population mean miles. Show all work.  Pooled standard error = nullnullnull.nullnull null nullnull null null nullnull = 197.3(0.2488) = 49.09 So the 95% CI is given by: (3120 ? 3050) ħ (2.00)(49.09) ? 70 ħ 98.19 ? (?20.18, 168.18) Final Answer: ( __?28.18 __ , __168.18__ ) c. Which of the following are correct interpretations of the 95% confidence level?  Circle all that apply. ? With repeated random samples, µ 1 ? µ 2 should fall in the interval computed in part (b) 95% of the time. ? With repeated random samples, µ 1 ? µ 2 should fall in 95% of the resulting intervals. ? There is a 95% chance that µ 1 ? µ 2 will fall in the interval computed in part (b). d. Based on the confidence interval, what would your decision be about testing the following hypotheses: H 0 : µ 1 = µ 2 versus H a : µ 1 ? µ 2 at a 5% significance level?  Circle one: Fail to Reject H 0 Reject H 0 Not Enough Information to Decide Explain briefly: Since the 95% confidence interval includes the value of 0, we cannot refute the statement that µ 1 ? µ 2 = 0. Stats 350 F09 Final Exam Solutions Page 5 of 10 Stats 350 Fall 2009 Final Exam Solutions 7. Car weight and Fuel Efficiency: A researcher wants to examine and model the relationship between the weight of a car and its fuel efficiency. To investigate this relationship, the data on the fuel efficiency (miles per gallon) and the weight (in 1,000lb) for 38 randomly selected cars were collected and a model was developed using linear regression. Use the plot and SPSS output below to answer the following questions. Stats 350 F09 Final Exam Solutions Page 6 of 10 Stats 350 Fall 2009 Final Exam Solutions Problem 7 continued a. Consider the following statements and clearly circle all that are correct:  ? With each 1lb increase in car weight, we would expect to see a decrease of 5.378 miles per gallon in fuel efficiency. ? With each 2,000lb increase in car weight, we would expect to see a decrease of 10.756 miles per gallon in fuel efficiency. ? Based on the analysis above, about 57.4% of the variation in fuel efficiency can be explained by the linear relationship with car weight. ? Based on the analysis above, about 33.0% of the variation in fuel efficiency can be explained by the linear relationship with car weight. b. Give the correlation between fuel efficiency and weight.  Final Answer: __ ? 0.574__ c. Give the equation of the estimated least squares regression line for predicting fuel efficiency from weight.  Final Answer: __ Predicted miles per gallon = 41.764 ? 5.378(weight) __ d. The unstandardized coefficient value of 41.764 given in the SPSS output is an example of:  Circle all that are correct: a sample statistic a population parameter a test statistic a slope of the regression line for the population an intercept of the regression line for the sample e. The actual fuel efficiency of the car that weighs 4,054lb in the data set was 15.5 miles per gallon. Compute the residual for this observation. Show all work and include the units.  Predicted miles per gallon = 41.764 ? 5.378(4.054) = 41.764 ? 21.802 = 19.962 residual = observed y ? predicted y = 15.5 ? 19.962 = -4.462 Final Answer: ______-4.462 mpg ____ Stats 350 F09 Final Exam Solutions Page 7 of 10 Stats 350 Fall 2009 Final Exam Solutions Problem 7 continued f. Explain in one simple sentence why it would not be appropriate to use the least squares regression line in part (b) to estimate the fuel efficiency for a car that weighs 6,000lb.  This would be extrapolation; it is not appropriate to predict for a value of x that is outside the range of the actual data used to develop the estimated model. g. The researcher would like to test if there is a significant negative linear relationship between fuel efficiency and car weights. Write out the appropriate hypotheses, the value of the test statistic, and the corresponding p?value. Then circle the appropriate conclusion using a 5% significance level.  H 0 : ____? 1 = 0________________ H a : _____? 1 < 0________________ Test statistic value = ____?4.210______ p?value = ___0.0002/2 = 0.0001___ Therefore, it appears that (circle one): ? heavier cars tend to have lower fuel efficiency. ? lighter cars tend to have lower fuel efficiency. ? there is not significant linear relationship between car weight and fuel efficiency. h. The 95% confidence interval for the average fuel efficiency for all cars that weigh 3,000lb would be  (circle one) narrower wider than the 95% confidence interval for the average fuel efficiency for all cars that weigh 4,000lb. i. At the right is a plot from the regression analysis above. Consider the following sentences and circle the appropriate one.  The plot at the right is used to check the assumption that: (circle one) ? The car weights are randomly distributed. ? The fuel efficiency values are randomly distributed. ? The true error terms have constant variance. ? The residuals have constant variance. Stats 350 F09 Final Exam Solutions Page 8 of 10 Stats 350 Fall 2009 Final Exam Solutions 8. Which classroom environment is best for teaching college pre?calculus? A total of 75 students were randomly assigned to one of three classes. The first class (group 1) emphasized group work. The second class (group 2) was a traditional lecture. The third class (group 3) emphasized individual instruction and practice. At the end of the year, students? knowledge was assessed with a standardized test. A school district wants to de ermine if the instruction method data was analyzed and some output is provided. State the h t has an effect on student outcomes, on average. The a. ypotheses to be tested. ] is different_ b. There are 4 missing values in the ANOVA table. Find the values and enter them clearly in the table. ] ution of the test statistic for testing the hypotheses in part (a)? Final Answer: __F (2, 72) distribution __ One of the assumptions in ANOVA is that all population variance common population standard deviation? Final Answer: __square root of 143.570 = 11.982__ e. At the 10% significance level, the decision is: (circle o [2 H 0 : _ ? 1 = ? 2 = ? 3 ____________ H a : _at least one population mean [2 c. If there was no difference in the average scores for the three instruction methods, what is the distrib  d. s are equal. What is the estimate of the  ne) Reject H 0 Fail to Reject H 0  re ractice method used by group 3. Provide the superintendant a 90% confidence interval estimate for the population  te: t* is based on 72 df or from the table using 70 df. .04__ , __84.04__ ) f. The school district superintendant would like to know mo about the individualized instruction and p mean test score for students taught using the individualized instruction and practice method. Show your work. 80.04 ħ t*(s p /?25) ? 80.04 ħ 1.67(11.982/5) ? 80.04 ħ 4.00 ? (76.04, 84.04) No Final Answer: ( __76 ANOVA Score Source Sum of Squares df Mean Square F Sig. Between Groups ____________ 2 416.813 __________ .061 Within Groups 10337.040 ______ 143.570 Total 11170.667 ______ Descriptives Score Group N Mean Std. Deviation Std. Error 1 25 71.8800 12.70276 2.54055 2 25 75.6800 12.52903 2.50581 3 25 80.0400 10.60063 2.12013 Total 75 75.8667 12.28637 1.41871 Stats 350 F09 Final Exam Solutions Page 9 of 10 Stats 350 Fall 2009 Final Exam Solutions Stats 350 F09 Final Exam Solutions Page 10 of 10 9. Absenteeism Study: The personnel director of a corporation ical a. ] ill be the midpoint of the interval, so the value of 9.5; unites here are days. .5 days___ b. Write a simple sentence to provide an interpretation of the interval (8, 11 We are 95% confidence that the population mean number of days absent is between 8 and 11 days. r wishes to study absenteeism among cler workers at the central office. A random sample of 25 clerical workers was obtained and the number of days absent over the previous year time frame was recorded. The director comes to you with the 95% confidence interval for the population mean number of days absent for all such clerical workers of (8, 11). The confidence interval estimate is provided above. What is the value of the sample mean? Include your units. [2 The sample mean w Final Answer: ___9 ).  O We would estimate (with 95% confidence) the population mean number of days absent to be somewhere between 8 and 11 days. 10. Name that Scenario: One important aspect in Statistics is to understand which statistical methods or dividual sale at _D__ 1. Is the average number of chocolate chips in our chippy e A. Simple linear regression B. 1?sample t?test for C. Paired t?test for a population D. 2?sample t?test for the E. 1?sample Z?test for a F. 2?sample Z?test for the ons H. Chi?squared test of I. Chi?squared test of procedures are appropriate to use to address the research problem or question of interest. The Janus Sisters make cupcakes and are considering providing their specialty treats for in local coffee shops. Before they embark on this endeavor, they have many plans to formulate and decisions to make. They need your help in deciding which statistical method to use to address their questions. For each question, select the letter corresponding to the statistical analysis technique most appropriate for addressing that question. [2 points each] _ cupcakes is higher than the average number of chocolat chips in our competitor?s chocolate chip cupcake? a population mean mean difference comparison of two population means population proportion comparison of two population proporti independence homogeneity __ E __ 2. Do a majority of consumers prefer the Janus cupcakes over our leading competitor?s cupcakes? __ H __ 3. Is there an association between the type of flour used ? (all?purpose flour, whole?wheat flour, cake flour) and the crumbliness of the cupcake (crumbly, not crumbly) __ A __ 4. How does the baking temperature affect the final moisture level of the cupcake? __ B__ 5. Is the average number of calories for our death by chocolate cupcakes more than 280? Lori
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