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INFERENCE REVIEW WORKSHEET #1 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) How many degrees of freedom are there for a chi -square test of independence based on a table with five rows and six columns? A) 24 B) 4 C) 5 D) 20 E) 30 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 2) Carnivores A random sample of some of the heaviest carnivores on Earth was reviewed to determine if there is an association between the length (in meters) and weight (in kilograms) of these carnivores. Here are the scatterplot, the residuals plot, a histogram of the residuals, and the regression analysis of the data. Use this information to analyze the association between the length and weight of these carnivores.
a. Is there an association? Write appropriate hypotheses. b. Are the assumptions for regression satisfied? Explain. c. What do you conclude? d. Create a 98% confidence interval for the true slope. e. Explain in context what your interval means. 3) Peanut M&Ms According to the Mars Candy Company, peanut M&Mʹs are 12% brown, 15% yellow, 12% red, 23% blue, 23% orange, and 15% green. On a Saturday when you have run out of statistics homework, you decide to test this claim. You purchase a medium bag of peanut M&Mʹs and find 39 browns, 44 yellows, 36 red, 78 blue, 73 orange, and 48 greens. Test an appropriate hypothesis and state your conclusion.
4) Voter registration A random sample of 337 college students was asked whether or not they were registered to vote. We wonder if there is an association between a studentʹs sex and whether the student is registered to vote. The data are provided in the table below (expected counts are in parentheses). (All the conditions are satisfied - donʹt worry about checking them.)
The calculated statistic is χ 2 = 0.249. a. Write appropriate hypotheses. b. Suppose the expected values had not been given. Show exactly how to calculate the expected number of men who are registered to vote. c. Show how to calculate the component of χ 2 for the first cell. d. How many degrees of freedom are there? e. Find the P-value for this test. f. State your complete conclusion in context. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 5) How many degrees of freedom are there for regression inference with 28 data values? A) 27 B) 56 C) 26 D) 54 E) 28
6) We randomly divide 200 volunteers with headaches into two groups who take identical -looking pills. One group gets a homeopathic remedy and the other a placebo. After 20 minutes we ask them to rate their headache pain as ʺno changeʺ, ʺsomewhat betterʺ, ʺmuch betterʺ, or ʺgoneʺ. What is the appropriate test? A) χ 2 test of homogeneity
B) χ 2 test of independence C) 2-proportion z-test D) 2-sample t-test E) matched pairs t-test 7) Itʹs common for a movieʹs ticket sales to open high for the first couple of weeks, then gradually taper off as time passes. Hoping to be able to better understand how quickly sales decline, an industry analyst keeps track of box office revenues for a new film over its first 20 weeks. What inference method might provide useful insight? A) t-test for linear regression B) t-Interval for slope C) 1-proportion z-test D) t-Interval for a mean E) χ 2 goodness-of-fit test
8) Several volunteers engage in a special exercise program intended to lower their blood pressure. We measure each personʹs initial blood pressure, lead them through the exercises daily for a month, then check blood pressures again. To see if the program lowered blood pressure significantly we should do a A) matched pairs t-test B) χ 2 test of homogeneity
C) linear regression t-test D) χ 2 goodness-of-fit test E) 2-sample t-test 9) Suppose that after the study described in #5 we want to see if thereʹs evidence that the exercise programʹs effectiveness in lowering blood pressure depends on how high the personʹs initial blood pressure was. We should do a A) 2-sample t-test B) matched pairs t-test C) χ 2 test of independence D) linear regression t-test E) χ 2 goodness-of-fit test
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) Height and weight Is the height of a man related to his weight? The regression analysis from a sample of 26 men is shown. (Show work. Donʹt write hypotheses. Assume the assumptions for inference were satisfied.)
a. How many degrees of freedom? b. What is the value of the t statistic? c. What is the P-value? d. State your conclusion in context.
Answer Key Testname: INFERENCE REVIEW WORKSHEET 1
1) D 2) a. H0 : There is no association between the length and weight of these carnivores. HA: There is an association between the length and weight of these carnivores. b. Conditions: *Straight enough: There is no obvious bend in the scatterplot of the data or in the plot of residuals against predicted values. *Independence: The residuals look random. *Does the plot thicken?: The residuals plot shows no disturbing changes in the spread about the line. *Nearly Normal: A histogram of the residuals does not deviate too much from being unimodal and symmetric. c. The P-value is very small, so we reject the null hypothesis. There is strong evidence of an association between carnivore length and weight. d. A 98% confidence interval for β1 is: β1 ± t*7 × SE(b1 ) = 485.21 ± 2.998(90.63) or (213.5, 756.9) kilograms. e. We are 98% confident that the weight of a heavy carnivore will be higher, on average, between 213.5 and 756.9 kilograms for each additional meter in length. 3) We want to know if the distribution of colors in the bag matches the distribution stated by the Mars Candy Company. H0 : The distribution of colors in the bag matches the distribution stated by the Mars Candy Company. HA: The distribution of colors in the bag does not match the distribution stated by the Mars Candy Company. Conditions: *Counted data: We have the counts of the number of peanut M&Ms of each color. *Randomization: We will assume that each bag of peanut M&Ms represents a random sample of peanut M&Ms. *Expected cell frequency: There are a total of 318 peanut M&Ms. The smallest percentage of any particular color is 12% (brown and red), and we expect 318(0.12) = 38.16. Since the smallest expected count exceeds 5, all expected counts will exceed 5, so the condition is satisfied. Under these conditions, the sampling distribution of the test statistic is χ 2 with 6 - 1 = 5 degrees of freedom, and we will perform a chi-square goodness-of-fit test.
(Obs - Exp)2 (39 - 38.16)2 (44 - 47.7)2 χ 2 = ∑ + +... = 0.7528 = 38.16 Exp 47.7 P-value = P(χ 2 > 0.7528) = 0.980 A P-value this large says that if the distribution of colors in the bag matches the distribution stated by the Mars Candy Company, an observed chi-square value of 0.7528 would happen about 98% of the time. Thus, we fail to reject the null hypothesis. These data do not show evidence that the distribution of colors in the bag does not match the distribution stated by the Mars Candy Company. 4) a. H0 : Voter registration is independent of a studentʹs sex. HA: There is an association between voter registration and a studentʹs sex. b. c.
251 137 = 102 337
(104 - 102)2 102
d. df = (2 - 1)(2 - 1) = 1 e. 0.618 f. Since the P-value of 0.618 is high, we fail to reject the null hypothesis. There is no evidence of an association between a studentʹs sex and whether the student is registered to vote. 5
Answer Key Testname: INFERENCE REVIEW WORKSHEET 1
5) C 6) A 7) B 8) A 9) D 10) a. df = 26 - 2 = 24 8.737 b. t = = 6.659 1.312 c. The P-value is 2 × P (t*24 > 6.659) < 0.0001. d. The P-value is very small, so we reject the null hypothesis. There is strong evidence that, on average, menʹs weights are about 8.7 pounds higher for each additional inch in height.