MONT 107N – Understanding Randomness Final Examination May 11, 2010

Directions: Do all work in the blue exam booklet. You must show all work necessary to justify your answers. There are 200 possible points, distributed as indicated in the questions.

1. Short Answer.

(a) (10) How are the EV and SE for the sum of n draws with replacement from a box computed?

(b) (10) What does the Central Limit Theorem say?

(c) (10) What is a 99% confidence interval for an average?

(d) (10) What is the p-value of a test of significance?

(e) (10) When is a t-test for an average used instead of a z-test?

2. A statistician tosses a coin 100 times. His null hypothesis is that the coin is fair, and his alternative is that the coin is biased in favor of heads. The coin comes up heads 60 times For each of the following, say whether the statement is true or false, and explain.

(a) (10) If the coin was fair, there would be about a 3% chance of getting 60 or more heads in 100 tosses.

(b) (10) Given that 60 heads are observed, there is only about a 3% chance that the coin is fair.

3. A gambler is playing roulette. On each play, he bets $1 on a “split” (two adjacent numbers), which pays 17 to 1. That is, if one of his 2 numbers comes up, he will win $17 on a $1 bet. Otherwise, he loses the $1.

(a) (5) Construct a box model for the payoff on this bet. (Hint: Recall there are a total of 38 slots on a roulette wheel.)

(b) (10) Compute the average and SD of the box.

(c) (10) If the bet is played 100 times, compute the EV and the SE for the payoff.

(d) (10) What are the gambler’s chances of coming out ahead on 100 spins of the wheel?

4. An airline does a market research survey on travel patterns. It takes a simple random sample of 225 people and works out the 95% confidence interval for the average distance they traveled on vacation in the previous year. The computed interval extends from 488 to 592 miles. Say whether each statement below is true or false and give reasons. If there is not enough information given to decide, explain what other information you would need.

(a) (5) The average distance traveled was about 540 miles.

(b) (5) The SD of the distances traveled was about 200 miles.

(c) (5) The probability histogram for averages of samples of size 225 drawn from the population is probably close to (scaled, shifted) normal curve.

(d) (5) The probability histogram for the distances traveled by randomly chosen individuals in the population is close to a (scaled, shifted) normal curve.

(e) (5) If the sample size is increased to 450, the width of the confidence interval will always be about half as wide.

5. A simple random sample of 1000 persons is taken to determine the percentage of Democrats in a large population. It turns out that 543 of the people in the sample are Democrats.

(a) (10) Compute a 95% confidence interval for the percentage of Democrats in the population.

(b) (10) True or False and explain: There is about a 5% chance that the actual percentage of Democrats is outside the interval found in part A.

6. The National Assessment of Educational Progress tested 17-year-olds on mathematics in 1990 and again in 2004. In 1990, the average score was 305 and the SD was 34. In 2004, the average score was 307 and the SD was 27. In both years, simple random samples of size 1000 were used. Can the difference be explained as chance variation, or have average mathematics scores really gone up?

(a) (5) Should you use a one-sample or a two-sample z-test for this kind of question?

(b) (10) Formulate null and alternative hypotheses in terms of a box model. (Say how many boxes there are, how many tickets go in each, and what is on the tickets – scores or 0/1’s.)

(c) (10) Carry out the test, find the associated p-value, and discuss your results.

7. A New York Times/CBS poll conducted between April 5 and April 12, 2010 included the question “Do you approve or disapprove of the way your Representative in Congress is handling his or her job?” The poll was carried out in two stages. First a large national simple random sample was asked the question. The results were that 46% of respondents said they approved, 36% said they disapproved, and 18% said they had no opinion. Then the same question was asked of a separate sample of 881 people, all of whom had identified themselves as supporters of the “Tea Party” movement. The responses broke down like this:

Approve Disapprove No opinion Total
“Tea Party” Sample 352  432  97  881

(Data is adapted from the New York Times web site.) It seems from the numbers that “Tea Party” supporters may differ from the general population when it comes to their opinions concerning their Congressional Representatives. But is this a real difference, or could it just be a product of chance variation in the sampling process?

(a) (5) Formulate a null and alternative hypothesis for this question.

(b) (5) What significance test should you use to decide between these?

(c) (10) Carry out your test of signficance. What do you get for a test statistic and observed significance level (p-value)?

(d) (5) Does this data give evidence of a real difference between “Tea Party” supporters and the public as a whole? Explain.

Have a safe, enjoyable, and productive summer!