Elementary Statistics Sample Exam #3 1. We wish to see if, on average, traffic is moving at the posted speed limit of 65 miles per hour along a certain stretch of Interstate 70. On each of four randomly selected days, a randomly selected car is timed and the speed of the car is recorded. The observed speeds are 70, 65, 70, and 75 miles per hour. Assuming that speeds are normally distributed with mean µ, we test whether, on average, traffic is moving at 65 miles per hour, by testing the following hypotheses. H0 : µ = 65, versus HA : µ 6= 65 Based on these data, A. we would reject H0 at significance level 0.10, but not at 0.05. B. we would reject H0 at significance level 0.05, but not at 0.025. C. we would reject H0 at significance level 0.025, but not at 0.01. D. we would reject H0 at significance level 0.01. 2. The physician-recommended dosage of a new medication is 14 mg. Actual administered doses vary slightly from dose to dose and are normally distributed with mean µ. A representative of a medical review board wishes to see if there is any evidence that the mean dosage is more than recommended and so intends to test the hypotheses given below. H0 : µ = 14, versus HA : µ > 14 To do this, he selects 16 doses at random and determines the weight of each. He finds the sample mean to be x̄ = 14.12 mg and the sample standard deviation to be s = 0.24 mg. Based on these data, A. we would reject H0 at significance level 0.10, but not at 0.05. B. we would reject H0 at significance level 0.05, but not at 0.025. C. we would reject H0 at significance level 0.025, but not at 0.01. D. we would reject H0 at significance level 0.01. 3. The water diet requires one to drink two cups of water every half hour from when one gets up until one goes to bed, but otherwise allows one to eat whatever one likes. Four adult volunteers agree to test the diet. They are weighed prior to the diet and after six weeks on the diet. The weights (in pounds) are as follows. Page 1 of 11 Elementary Statistics Sample Exam #2 Person Weight before the diet Weight after six weeks 1 180 170 2 125 130 3 240 215 4 150 152 For the population of all adults, assume that the weight loss after six weeks on the diet (weight before beginning the diet minus weight after six weeks on the diet) is normally distributed with mean µ. To determine if the diet leads to weight loss, we test the following hypotheses. H0 : µ = 0 versus HA : µ > 0. Based on these data, A. we would not reject H0 at significance level 0.10. B. we would reject H0 at significance level 0.10, but not at 0.05. C. we would reject H0 at significance level 0.05, but not at 0.01. D. we would reject H0 at significance level 0.01. Use the following to answer question the next three problems: A new diet has been developed for raising beef cows. Two random samples of size nine are independently selected, and one is given the standard diet and the other is given the new diet. After 18 weeks, the weight gain was measured, and it was found that x̄1 = 30 with s1 = 8 and x̄2 = 26 with s2 = 6. Let µ1 and µ2 represent the mean weight gain we would observe for the entire population of beef cows when on, respectively, a new diet and a standard diet. Assume that two–sample t procedures are safe to use. 4. Suppose the researcher had wished to test the following hypotheses H0 : µ1 = µ2 versus HA : µ1 > µ2 . The numerical value of the two-sample t statistic is A. 0.36. B. 1.20. C. 2.57. D. 4.00. 5. Suppose the researcher had wished to test the following hypotheses. H0 : µ1 = µ2 versus HA : µ1 > µ2 . The P -value for the test is (use the conservative value for the degrees of freedom) Page 2 of 11 Elementary Statistics Sample Exam #2 A. larger than 0.10. B. between 0.10 and 0.05. C. between 0.05 and 0.01. D. below 0.01. 6. A 99% confidence interval for µ1 −µ2 is (use the conservative value for the degrees of freedom) A. 4 ± 4.7. B. 4 ± 6.2. C. 4 ± 7.7. D. 4 ± 11.2. Use the following to answer the next three questions: An inspector inspects a shipment of medications to determine the efficacy in terms of the proportion p in the shipment that failed to retain full potency after 60 days of production. Unless there is clear evidence that this proportion is less than 0.05, he will reject the shipment. To reach a decision, he will test the following hypotheses using the large sample test for a population proportion. H0 : p = 0.05 versus HA : p < 0.05. To do so, he selects an SRS of 200 pills. Suppose that eight of the pills have failed to retain their full potency. 7. The P -value of his test is A. 0.2352. B. 0.2582. C. 0.4704. D. 0.5164. 8. Which of the following assumptions for inference about a proportion using a hypothesis test are violated? A. The data are an SRS from the population of interest. B. The population is at least 10 times as large as the sample. C. n is so large that both np0 and n(1 − p0 ) are 10 or more, where p0 is the proportion with major defects if the null hypothesis is true. Page 3 of 11 Elementary Statistics Sample Exam #2 D. There appear to be no violations. 9. A 95% plus four confidence interval for the true proportion of pills in the shipment that have failed to retain full potency is A. 0.049 ± 0.0269. B. 0.049 ± 0.0272. C. 0.049 ± 0.0296. D. 0.049 ± 0.0299. Use the following to answer the following three questions: A phycologist is worried that that a large proportion of the algae found in nearby water sources is cyanobacteria. In particular, he obtains a random sample of 50 alga from a local river and an independent sample of size 75 from the local coastline. He finds that 22 of the river samples and 66 of the coastline samples are cyanobacteria. He considers the river to act as a control group. Let p1 and p2 represent the proportion of cyanobacteria in riparian and coastal populations, respectively. 10. Is there evidence that the proportion of cyanobacteria is higher in coastal samples than in riparian samples? To determine this, you test the following hypotheses. H0 : p1 = p2 versus HA : p1 < p2 . The value of the z statistic for testing these hypotheses is A. z = −5.98. B. z = −5.78. C. z = −5.58. D. z = −5.28. 11. Is there evidence that the proportion of cyanobacteria is higher in coastal samples than in riparian samples? To determine this, you test the hypotheses H0 : p1 = p2 versus HA : p1 < p2 . The P -value of your test is A. between 0.10 and 0.05. B. between 0.05 and 0.01. C. between 0.01 and 0.001. Page 4 of 11 Elementary Statistics Sample Exam #2 D. below 0.001. 12. A 99% confidence interval for p1 − p2 is A. −0.44 ± 0.0163. B. −0.44 ± 0.1896. C. −0.44 ± 0.2054. D. −0.44 ± 0.2149. Use the following to answer the next three questions: A physiologist is interested in determining the proportion of algae samples from a local rivulet that belong to a particular phyla, and he believes they should be uniformly distributed. A random sample of 60 alga were obtained, and each was categorized as being Rhodophyta, Chlorophyta, or Heterokontophyta. The observed counts were 25, 25, and 10, respectively. 13. The chi–square statistic is A. 0. B. 7.50. C. 20. D. 150. 14. When determining the significance of the chi–square statistic, the phycologist would use A. 1 degree of freedom. B. 2 degrees of freedom. C. 3 degrees of freedom. D. 4 degrees of freedom. 15. The significance level of the chi–square statistic in this case is A. greater than 0.10. B. below 0.10 but above 0.05. C. below 0.05 but above 0.01. D. below 0.01. Page 5 of 11 Elementary Statistics Sample Exam #2 Use the following to answer the following two questions: A fisheries biologist is interested in studying the relationship between width and weight in horseshoe crabs. She collects a random sample of such crabs and cross– classifies them based on these variables as given below. Weight < 1.8kg > 1.8kg 0–5 39 11 Width (in cm) 5–10 10–15 15–20 36 29 18 14 21 32 16. Suppose we wish to test the null hypothesis that there is no association between their width and weight. Under the null hypothesis, what is the expected number of crabs in the low-weight class and widest–width class? A. 18.0 B. 25.0 C. 30.5 D. 50.0 17. Which hypotheses are being tested by the chi–square test? A. The null hypothesis is that width and weight are independent, and the alternative is that they are dependent. B. The null hypothesis is that the mean number of crabs that are in the low weight-class is the same for each of the four width classes, and the alternative is that these means are different. C. The null hypothesis is that the distributions of the number of crabs that are in the low– and high–weight classes are the same for the four widths. The alternative says the distributions are different. D. The null hypothesis is that the distributions of the total number of crabs sampled in each of the four widths are the same. The alternative is that these distributions are different. 18. In a χ2 test for independence, the statistic based on a contingency table with 6 rows and 5 columns will have how many degrees of freedom? A. 30 B. 24 Page 6 of 11 Elementary Statistics Sample Exam #2 C. 5 D. 20 E. 25 19. The two variables in a scatter plot are called the A. independent variable and dependent variable. B. relative frequency and relative proportion. C. spread and pattern. D. lines and points. 20. A classic assumption in regression is that the values of Y at any value of X follow what kind of distribution? A. Binomial B. Normal C. Uniform D. Gamma 21. The line described by the regression equation attempts to A. pass through as many points as possible. B. pass through as few points as possible. C. minimize the number of points it touches. D. minimize the squared distance from the points. 22. A clinical psychologist finds the relationship between the number of weeks spent in a therapy hospital (X = HOSPITAL) and number of seizures per week (Y = SEIZURES) is described by the following equation: Ŷ = 14.09 − 0.91X. This is based on a sample size of 50 patients and is associated with r = −0.93. The proportion of variance in SEIZURES accounted for by HOSPITAL (i.e., the coefficient of determination) is A. 0.93 B. -0.93 C. 0.86 D. . -0.86 Page 7 of 11 Elementary Statistics Sample Exam #2 E. 14.09 23. Suppose a straight line is fit to data having response variable y and explanatory variable x. Predicting values of y for values of x outside the spread of the observed data is called A. contingency. B. extrapolation. C. causation. D. correlation. 24. Changing the units of measurements on the Y variable will affect all but which one of the following? A. The estimated intercept parameter. B. The estimated slope parameter. C. The total sum of squares for the regression. D. R squared for the regression. E. The estimated standard errors. 25. A researcher wishes to determine whether the rate of water flow (in liters per second) over an experimental soil bed can be used to predict the amount of soil washed away (in kilograms). The researcher measures the amount of soil washed away for various flow rates, and from these data calculates the least– squares regression line to be amount of eroded soil = 0.4 + 1.3 ∗ (flow rate). The correlation between amount of eroded soil and flow rate would be A. 1/1.3. B. 0.4. C. positive, but we cannot say what the exact value is. D. either positive or negative. It is impossible to say anything about the correlation from the information given. 26. Suppose we fit the least-squares regression line to a set of data. Points with unusually large values of the residuals are called A. response variables. B. the slope. Page 8 of 11 Elementary Statistics Sample Exam #2 C. outliers. D. correlated. For the following four questions use the following information. At what age do babies learn to crawl? Does it depend on the time of the year that babies are born? Data were collected from parents who brought their babies into the University of Denver’s Infant Study Center to participate in one of a number of experiments between 1988 and 1991. Parents reported the birth month and the age in which their child was first able to creep or crawl a distance of four feet within one minute. The resulting data were grouped by month of birth: January, May, and September. Birth month January May September Mean 29.84 28.58 33.83 Std. dev. 7.08 8.07 6.93 n 32 27 38 Crawling age is given in weeks. Assume that the data can be considered as three independent random samples, one from each of the populations comprised of babies born in that particular month, and that the populations of crawling ages have Normal distributions. An ANOVA was run on the data. The following shows a portion of the results. Source Group Error Total df Sum of squares 505.26 Mean square F –ratio 53.45 27. The numerator degrees of freedom for the ANOVA are A. 2. B. 3. C. 94. D. 96. 28. The mean square for groups is A. 5.38. Page 9 of 11 Elementary Statistics Sample Exam #2 B. 252.63. C. 1010.52. D. 1515.78. 29. The value of the F statistic is A. 3.15. B. 4.73. C. 6.30. D. 9.45 30. The p–value for the ANOVA that tests for equality of the population means of the three months is A. less than 0.01. B. between 0.010 and 0.025. C. between 0.025 and 0.05. D. greater than 0.05. Researchers are interested in the effects of wind direction and season on ozone levels in North America. The ozone levels were recorded at locations, along with the direction of prevailing winds (E, N, S, W) and the season (Winter, Spring, Summer, Fall). A balanced design was carried out such that each combination was observed five times. A partial ANOVA table is provided. Source DF SS MS Direction 129 Season 318 Interaction 423 Error 960 Total 1,830 31. The mean square for the interaction is A. 3.13. B. 47. C. 70.5. D. 423. Page 10 of 11 F Elementary Statistics Sample Exam #2 32. The F statistic for the interaction is A. 3.13. B. 4.70. C. 47. D. 423. 33. Based on the value of the F statistic for the interaction, and using a 0.05 significance level, one would conclude that A. the p–value is below 0.05, so a significant interaction is present. B. the p–value is above 0.05, so a significant interaction is present. C. the p–value is below 0.05, so a significant interaction is not present. D. the p–value is above 0.05, so a significant interaction is not present. Page 11 of 11