STAT 239 Exam 2 Name: 1. (30 points) A forced-choice design is often used to compare the attractiveness of pheromones to insects. A Y-tube is used. The pheromone is place on one branch; the control on the second branch; and the insect on the third branch. The insect then chooses one of the two branches. Suppose that 45 of 75 insects choose the pheromone branch. The 95% large-sample confidence interval for the proportion of the population that prefers the pheromone branch is: (a) 0.60 0.06. (b) 0.60 0.11. (c) 0.60 0.006. (d) 0.60 0.14. The null and alternative hypotheses for testing the hypothesis of no preference are: (a) (b) (c) (d) What is the value of the test statistic for testing the hypothesis of no-preference? (a) 1.73 (b) 1.76 (c) 10.60 (d) 10.40 What is the p-value for the hypothesis test of no preference? (a) 0.64 (b) 0.05 (c) 0.015 (d) 0.08 2. (30 points) A drug manufacturer wishes to compare the incidence of side-effects from two different formulations of a drug. In a sample of 400 patients using formulation 1, there are 20 patients who experience side-effects. In a sample of 100 patients using formulation 2, there are 10 patients who experience side-effects. A 95% confidence interval for the difference (formulation 1 - formulation 2) in side-effect rates is: (a) -0.05 0.068. (b) -0.05 0.063. (c) -0.05 0.033. (d) 0.05 0.038. The null and alternative hypothesis to test no difference in side-effects rates between the two formulations is: (a) (b) (c) (d) Note: H and A are null and alternative hypothesis. Find the value of the test statistic and the p-value for the test of no difference in sideeffect rates between the two formulations. 3. (30 points) Does taking Vitamin D increase the mean lifespan of a rat? A sample of 20 rats was randomized to group 1 (control) and group 2 (Vitamin D). The lifespan (days) of each rat was measured. Let 1 and 2 denote the mean life span of all rats that take the control and Vitamin D, respectively. 1) Based on software, the 95% confidence interval of the mean difference 1 - 2 is from 0.66 to 160.66. Interpret the result. 2) If we want to conduct a test for the claim that Vitamin D increases the mean life span of rats, specify the appropriate hypotheses. The p-value of the test is 0.0259, what conclusion will you draw at level significance level 0.05? 4. (35 points) Can rats learn? The following table is the time(s) taken for rats to run a maze twice: Rat 1 2 3 4 First time 440 600 720 630 Second time 450 520 720 600 Changes Assume that the changes in time (second time - first time) have an approximately Normal distribution. Carry out an appropriate matched pairs t test for the problem, following the standard 4-step procedure. STAT 239 Homework #4 (Due: 4/27) (Chi-square Tests, Inference for Regression, and One-Way Analysis of Variance) 1 of 6 Resource-selection analysis compares the distributions of animals relative to the distribution of habitat. If the two don't agree, there is evidence of selection. A survey of 106 moose found that 24 were located in "In burn interior," 22 in "In burn - edge," 20 in "Out of burn - edge," and 40 in "Other" habitat types. The corresponding proportion of the landscape as determined by a Geographic Information System (GIS) in these habitats was 0.340, 0.101, 0.104, and 0.455, respectively. The null hypothesis is: a) b) c) d) The expected number of moose in the "other" habitat is: a) 40.0. b) 52.8. c) 34.4. d) 48.2 The overall chi-square test statistic is: a) 44.83. b) 24.65. c) 36.35. d) 7.49. The degrees of freedom for the chi-square test statistics is: a) 1. b) 2. c) 3. d) 4. The p-value for the goodness-of-fit test is <0.0001. Which of the following is correct? a) There is less than a 1% chance that the moose have expressed any selection. b) If the moose do not express a selection for habitat, there is less than a 1% chance of seeing these observed locations. c) There is less than a 1% chance that the moose have not expressed any selection. d) If the moose express resource selection, then there is less than a 1% chance we would observe this data. 1|Page 2 of 6 A random sample of 140 births from local records showed this distribution across days of the week: Day Sun Mon Tue Wed Thurs Fri Sat Births 13 23 24 20 27 18 15 Here are the results of an analysis of this data to examine if the distribution is uniform: Which of the following is correct? a) The null hypothesis is that the proportion of births varies by day of the week. b) The chi-square test-statistic computed in this chapter has the value 7.76. c) There is very strong evidence that the distribution of births is not uniform across the week. d) The expected number of births on a Sunday under the null hypothesis is 20. 3 of 6 Are different species of mosquitoes attracted differently to the two sexes? A study was conducted where three closely related species of mosquitoes were presented with an arm from a male or a female and the selection was recorded. Here is the two-way table of counts: Female Male Species A 248 66 Species B 514 105 Species C 351 225 What proportion of mosquitoes of Species A chose the male arm? a) 4.4% b) 16.7% c) 21.0% d) 20.8% What is the null hypothesis of interest? a) Mosquitoes prefer female arms. b) There is no association between the species of mosquitoes and the arm selected. c) The proportion of mosquitoes in the three species is equal. d) The proportion of mosquitoes that prefer female arms is higher than the proportion that prefer male arms. 2|Page The expected number of mosquitoes of Species A who prefer male arms under the null hypothesis is approximately: a) 82.4. b) 16.7. c) 21.0. d) 4.4. The degrees of freedom for the a) 65. b) 6. c) 1. d) 2. test for the preceding table is: The results of the statistical test for the preceding table include a p-value of <0.001: Your conclusion is: a) There is no evidence of a relationship between the species of mosquito and the sex of the arm. b) There is evidence that mosquitoes target women. c) There is evidence that the numbers of the three species that prefer female arms is different. d) There is evidence that the three species differ in their preferences for the two types of arms. Which of the following is correct? a) The chi-square test may not be completely appropriate because the smallest cell count is less than 100. b) The chi-square test may not be completely appropriate because each arm has several bites. c) The chi-square test may not be completely appropriate because all species bite both types of arms. d) The chi-square test may not be completely appropriate because the table is not a 2 x 2 table. 4 of 6 One possible effect of air pollution is genetic damage. A study exposed a group of mice to air near a steel mill and another group of mice to air in a rural area to see if the mutation rate was higher for mice exposed to steel mill air. The number of mice that had mutations in their offspring was recorded: No Mutation Mutation present Rural 150 23 Steel 66 30 Mill The alternative hypothesis is: a) The rate of mutations is equal for mice exposed to rural and steel mill air. b) The rate of mutations is higher for mice exposed to steel mill air compared to rural air. c) The rate of mutations is different between mice exposed to steel mill air compared to rural air. d) There is no association between the mutation rate and the type of air mice were exposed to. The expected number of mice with mutations exposed to steel mill air is: a) 30.0. b) 18.9. c) 34.1. 3|Page d) 77.1. The chi-square test statistic was 12.6 and the p-value was < 0.001. Which of the following is correct? a) Because the p-value is 12.6%, there is no evidence against the hypothesis. b) This chi-square value is compared to a chi-square distribution with 4 degrees of freedom. c) There is strong evidence against the hypothesis of no association at significance level 0.05. The approximate relative risk of mutation in steel mill air compared to rural air is: (The relative risk is defined as the ratio of rate (proportion) of mutation for mice exposed to steel mill air and rate (proportion) of mutation for rural air.) a) 2.4. b) 0.42. c) 0.31. d) 1.3. 5 of 6 Blood pressure tends to increase with weight. A sample of 14 male employees who worked at a local business had their systolic blood pressure measured (mm Hg) along with their body weight (kg). Here is some information about the fit of the linear model: Which of the following is correct? a) The blood pressure is estimated to increase 1.65 mm for every additional kg of weight. b) The blood pressure is estimated to increase 6.39 mm for every additional kg of weight. c) The blood pressure is estimated to increase 7.05 mm for every additional kg of weight. d) The price is estimated to increase 0.23 mm for every additional kg of weight. Which of the following is the fitted regression line? a) Weight = 6.39 + 1.65(Blood Pressure) b) Weight = 1.65 c) Blood Pressure = 6.39 + 1.65(Weight) d) Blood Pressure = 1.65(intercept) + 0.23(Weight) Which of the following is correct? a) Ninety-five percent of men have blood pressure between 1.14 and 2.17 mm. b) The youngest man had a blood pressure of 6.38. c) The estimated increase in blood pressure for each additional kilogram of mass is between about 1.13 mm and 2.17 mm. d) The estimated increase in blood pressure for the heaviest man is less than about 45.2 mm for each additional kilogram of mass. 4|Page Here is some information about the fit of the linear model: Which of the following is correct? a) The mean blood pressure is about 33.85. b) The correlation between blood pressure and body mass is about 0.43. c) The standard deviation about the regression line is about 33.8 mm. d) The margin of error is about 33.8 mm. Here is some information about the fit of the linear model: The estimated blood pressure for a man of 80 kg is: a) 138.4 mm. b) 132.0 mm. c) 510.4 mm. d) 125.0 mm. Here is a plot of the regression line (the solid one, the dashed ones a give prediction band): The residual for the point indicated by a circle is about: a) 155. b) 86. c) 148. d) 7. Here is some information about the fit of the linear model: 5|Page The null hypothesis tested by the line labeled "Weight (kg)" is: a) The men do not vary in their weight. b) The average blood pressure does not depend on the body weight. c) The average man has a blood pressure that does not depend on the body weight. d) The estimated change in body weight for each change in blood pressure is 1.65 kg/mm. 6 of 6 Many studies have suggested a link between exercise and healthy bones because exercise stresses the bones and this causes them to become stronger. A study randomized subjects to a control (no additional exercise), low impact, and high impact aerobics and measured the change in bone density. The following is the output from an analysis of the experiment: The null and alternative hypotheses for this experiment are: a) b) c) d) Which of the following is correct? a) There is less than a 1% chance that the means are equal. b) The larger mean is 7.97 times likely to be different than the smaller mean. c) There is strong evidence that sample means could be different. d) If the population means were equal, the observed data are highly unusual. Which of the following is correct? a) The sample standard deviations are too unequal, so the results of the ANOVA cannot be trusted. b) The mean square error is an estimate of the within group standard error. c) About 95% of individuals in the control group had a change in bone density between about 580 and 620. d) Based on the overlap of the confidence intervals, there is little evidence to separate the means of the low impact group and the control group. 6|Page The following is a side-by-side boxplot of the individual scores: Which of the following is correct? a) We are 95% confident that the mean score of the control group is between about 550 and 650. b) About 25% of individuals in the LowJump group have scores above 600. c) About 95% of individuals in the HighJump group have scores that are above the LowJump group. d) The median score of the Control and LowJump group appear to be about equal. The ANOVA p-value was 0.023. Here are the results of a multiple comparison method following an ANOVA: Which of the following is correct? a) There is a 2.3% chance that the means are different. b) There is insufficient evidence to distinguish the population means of the Control and LowJump groups. c) There is a 2.3% chance that the HighJump mean is different from the other two means. d) There is no evidence that the mean for the HighJump group may be different from the other means. The ANOVA p-value was 0.023. Which of the following is correct? a) The significant p-value only indicates that there is evidence that at least one pair of means are different. It doesn't indicate which of the means are different. b) A multiple comparison procedure is needed because the p-value indicates that the means could all be the same. c) The Analysis of Variance (ANOVA) method tests to see if the variances in the groups could be the same. d) The results could be suspect because 30 people represents too small of a sample to ensure that the results are representative. The study concluded in its report, "There was strong evidence from the ANOVA that the scores under highimpact aerobics were larger than the scores in the two other groups (p = 0.023)." Which of the following is correct? a) There was a 2.3% difference in mean scores among the groups. b) The conclusion is wrong because ANOVA tests for equality of means and not individual scores. c) About 2.3% of individuals had lower scores under the high-impact regime than the control group. d) The conclusion is wrong because Analysis of Variance (ANOVA) method compares the variance of the individual scores. 7|Page The estimate of the within group standard deviation is about: a) 7.97. b) 460. c) 12,000. d) 22. The following is a plot of the residuals from an analysis of the experiment: Which of the following is correct? a) The distribution of residuals appears to be skewed, which violates one of the assumptions of ANOVA. b) The assumption of Normality is approximately correct because all of the residuals are between -60 and +60. c) The potential outlier between 50 and 60 in the residuals implies that the results of the ANOVA are suspect. d) The standard deviation within the groups is about 20. Which of the following is correct? a) ANOVA is preferred over the pairwise t tests because the computations are simpler. b) ANOVA is preferred over the pairwise t tests because it controls for the problem of multiple comparisons. c) The ANOVA hypotheses are that the means are all equal versus that the means are all different. d) The ANOVA hypotheses are that the means and standard deviations are all about equal. 8|Page