AP Statistics Final Examination Free-Response Questions

Transcription

1 AP Statistics Final Examination Free-Response Questions Name Date Period Section II Part A Questions 1 4 Spend about 50 minutes on this part of the exam (70 points) Directions: You must show all work and indicate the methods you use. You will be graded on the correctness of your methods and on the accuracy of your results and explanations. Go to the next page

2 1. A manufacturer of boots plans to conduct an experiment to compare a new method of waterproofing to the current method. The apperance of the boots is not changed by either method. The company recruits 100 volunteers in Seattle, where it rains frequently, to wear the boots as they normally would for 6 months. At the end of the 6 months, the boots well be returned to the company to be evaluated for water damage. (a) (10 points) Describe a design for this experiment that uses the 100 volunteers. Include a few sentences on how it would be implemented. (b) (5 points) Could your design be double blind? Explain.

3 2. Every Monday a local radio station gives coupons away to 50 people who correctly answer a question about a news fact from the previous day s newspaper. The coupons given away are numbered from 1 to 50, with the first person receiving coupon 1, the second person receiving coupon 2, and so on, until all 50 coupons are given away. On the following Saturday, the radio station randomly draws numbers from 1 to 50 and awards cash prizes to the holders of the coupons with these numbers. Numbers continue to be drawn without replacement until the total amount awarded first equals or exceeds $300. If selected, coupons 1 through 5 each have a cash value of $200, coupons 6 through 20 each have a cash value of $100, and coupons 21 through 50 each have a cash value of $50. (a) (5 points) Explain how you would conduct a simulation using the random number table provided below to estimate the distrubution of the number of prize winners each week (b) (10 points) Perform your simulation 3 times. (That is, run 3 trials of your simulation.) Start at the leftmost digit in the first row of the table and move across. Make your procedure clear so that someone can follow what you did. You must do this by marking directly on or above the table. Report the number of winners in each of your 3 trials.

4 3. The Earth s Moon has many impact craters that were created when the inner solar system was subjected to heavy bombardment of small celestial bodies. Scientists studied 11 impact craters on the Moon to determine whether there was any relationship between the age of the craters (based on radioactive dating of lunar rocks) and the impact rate (as deduced from the density of the craters). The data are displayed in the scatterplot below. Impact Rate grams/million years 8,000 6,000 4,000 2, Age (billions of years) (a) (5 points) Describe the nature of the relationship between impact rate and age. Go to the next page

5 (Problem #3, continued) Prior to fitting a linear regression model, the researchers transformed both impact rate and age by using logarithms. The following computer output and residual plot were produced. Regression Equation: ln(rate) = ln(age) Predictor Coef SE Coef T P Constant ln(age) S = R-Sq = 89.4% R-sq (adj) = 88.2 % Impact Rate grams/million years Age (billions of years) 8.5 (b) (7 points) Interpret the value of r 2 (c) (8 points) Comment on the appropriateness of this linear regression for modeling the relationship betwen the transformed variables.

6 4. The principal at Crest Middle School, which enrolls only sixth-grade students and seventh-grade students is interested in determining how much time students at that school spend on homework each night. The table below shows the mean and standard deviation of the amount of time spent on homework each night (in minutes) for a random sample of 20 sixth-grade students and a separate random sample of 20 seventh-grade students at this school. Sixth-grade students Seventh-grade students Mean Standard Deviation Based on dotplots of these data, it is not unreasonable to assume that the distribution of times for each grade were approximately normally distributed. (a) (10 points) Estimate the difference in mean times spent on homework for all sixth- and seventh-grade students in this school using an interval. Be sure to interpret your interval. (b) (10 points) An assistant principal reasoned that a much narrower confidence interval could be obtained if the students were paired based on their responses; for example, pairing the sixth-grade student and seventh-grade student with the next highest number of minutes spent on homework, and so on. Is the assistant principal correct in thinking that matching students in this way and then computing a matched-pairs confidence interval for the mean difference in time spent on homework is a better procedure than the one used in part (a)? Explain why or why not.

7 Section II Part B Question 5 Spend about 25 minutes on this part of the exam (30 points) Directions: You must show all work and indicate the methods you use. You will be graded on the correctness of your methods and on the accuracy of your results and explanations. Go to the next page

8 5. One of your friends knows that you are taking AP Statistics and asks for your help with her chemistry lab report. She has come up with five measurements of the melting point of a compound: 122.4, 121.8, 122.0, 123.0, and degrees Celsius. (a) (6 points) The lab manual asks for a 95% confidence interval estimate for the melting point. Show her how to find this estimate. (b) (6 points) Explain to your friend in simple language what 95% confidence means. (c) (6 points) Would a 90% confidence inteval be narrower or wider than the 95% interval? Explain your answer to your friend. (d) (6 points) The lab manual asks whether the data show sufficient evidence to call into question the established melting point of 122 degrees Celsuis. State the null and alternative hypotheses and find the P -value. (e) (6 points) Explain the meaning of the specific P -value to your friend.

9 Formulas I. Decriptive Statistics xi x = n 1 s x = (x1 x) n 1 2 (n 1 1)s (n 2 1)s 2 2 s p = (n 1 1) + (n 2 1) ŷ = b 0 + b 1 x (x1 x)(y i y) b 1 = (sx x) 2 b 0 = y b 1 x 1 ( ) ( ) x 1 x yi y r = n 1 b 1 = r s y s x (yi y) 2 s x s y s b1 = n 2 (xi x) 2 II. Probability P (A B) P (A B) = P (B) P (A B) P (A B) = P (B) E(X) = µ X = x i p i Var(X) = σx 2 = (x i µ X ) 2 p i If X has a binomial distribution with parameters n and p, then: P (X = k) = ( ) n k p k (1 p) n k µ X = np σ X = np (1 p) µˆp = p p(1 p) σˆp = n If x is the mean of a random sample of size n from an infinite population with mean µ and standard deviation σ, the: µ x = µ σ x = σ n

10 III. Inferential Statistics Standardized test statistic: Confidence interval: statistic parameter standard deviation of statistic statistic ± (critical value) (standard deviation of statistic) Statistic Sample Mean Sample Proportion Single-Sample Standard Deviation of Statistic σ n p (1 p) n Statistic Difference of sample means Difference of sample proportions Two-Sample Standard Deviation of Statistic σ 2 1 n 1 + σ2 2 n 2 Special case when σ 1 = σ 2 1 σ + 1 n 1 n 2 p 1 (1 p 1 ) + p 2 (1 p 2 ) n 1 n 2 Special case when p 1 = p 2 = p p (1 p) 1 n n 2 Chi square test statistic = (observed = expected) 2 expected

11 Table entry Table entry for z is the area under the standard normal curve left of z. z TABLE A z Standard normal probabilities

12 5 Table entry! " # $ % % & ' ( ) * + (, - '. % " ( % " / & 0 % ( '. % - ' " & 0 " ( 0 & + ( 1 " $ 2 / ( 3 % ' + '. % $ % * ' + * 5 4 z TABLEA St c z.

13 Probability p Table entry for p and C is the critical value t with probability p lying to its right and probability C lying between t and t. t* TABLE C t distribution critical values Upper tail probability p df z % 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% Confidence level C

14 Table entry for p is the critical value x with probability p lying to its right. x* Probability p TABLE E Chi-square distribution critical values df p