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STATISTICS X10

FINAL

Version

Coronavirus

1. A certain town has 9,000 families.

Population average mileage driven per family is 15,000 miles per year and the population SD is 2,000 miles per year. Fifteen percent of these families have no cars at all.

As part of an opinion survey, a simple random sample of 900 families (from this town) is chosen. What is the chance that sample average mileage driven per family is between 14,950 and 15,100 miles per year? Use the normal approximation method with continuity correction.

2. Define the following terms:

(a) population SD for 0-1 box =     (write formula)

(b) population SD =     (write formula)

(c) SD+ =     (write formula)

3. Assuming that a simple random sample of size 250 is being selected from a population that is approximately normally distributed (N=100,000). Assuming that the population SD is equal to 5 and the sample average is equal to 30, determine an 90% confidence interval for the true average of this population. (If this confidence interval can not be determined, explain why not.)

4. Somebody picks one ticket at random from a normal population (sample size is equal to 1). The ticket shows a “2”, i.e. the sample average is “2”. Assuming that the SD of the box is 3, determine the 95% confidence interval for the average of the box. (Remark: You should be able to construct a z or t confidence interval.)

5. Determine a 99% confidence interval for the population average of a normal distribution, given a random sample of size 12 with sample average = 2 and sample SD = 3.

6. (a) Determine the chance that the gambler wins when he bets on 4 or 11 when rolling two dice (in craps). Assume the dice are fair. (Hint: Draw a 6 by 6 grid and count the points where the sum of the dots is 4 or 11. How many out of 36?)

(b) A gambler is betting continuously on the event that the sum of the two dice in a craps game is 4 or 11. He wins in 120 of the next 900 games.

Determine a 95% confidence interval for the winning proportion for this gambler, i.e. for getting a 4 or 11 for the sum of two dice. Do not assume that the dice are fair and simplify your answer.

(c) Based on a comparison of the answers to part (a) and (b), do you believe that there was something wrong with the dice.

7. Given: A random sample of size 13, alpha=5%, sample average=7, sample SD=4, and population is normally distributed.

Determine the p-value in the following case and decide if you can reject H(0) at the indicated significance level of the test.

H(0): true average = 6.

H(1): true average > 6.

8. Assume that you have two zero-one boxes, one having a 0.50 proportion of ones and the other having a 0.7 proportion of ones. If we now select 25 tickets with replacement from each box, what is the SE for the difference in the proportion (averages) of “ones” obtained? (If it is not possible, explain why not? Otherwise, solve and not simplify your answer.)

9. John is in charge of the accounts receivable department of the LUVs company. An accountant wishes to check on the department and asks John for his ideas about outstanding accounts. John replies that he believes that the accounts should be distributed about as follows:

Less than 3 months delinquent    40%

3-6 months delinquent    30%

6-9 months delinquent    15%

More than 9 months delinquent 15%

The accountant, knowing that it is impractical to examine all of the accounts, chooses a random sample of 60 accounts and finds

20, 16, 12, and 12

accounts in these categories, respectively. Using a Chi-square Goodness of Fit Test decide if you can reject John’s hypothesis at a 5% level.

10.(a)The following data represents a random sample of patients in an U.S. mental hospital:

Test the null hypothesis that the method of the therapy is independent of the rating assigned against the alternative that the rating and the therapy are dependent. (Use alpha=5%, and state the p-value.)

(b) Repeat part (a) with the following data: