Normal Distribution

1. Suppose that the mean income of all Mudville residents is $40,000 with a standard deviation of $6,000. Answer the following questions in terms of these data.
a. What proportion of all possible samples of 36 Mudville residents would have a mean income of $38,000 or more?
b. What is the probability of selecting a random sample of 64 persons from this population and obtaining a sample with a mean income of between $39,000 and $40,000?

2. Suppose that 40% of all faculty at Penn State are women.
a. What is the probability of selecting a random sample of 50 Penn State faculty members and obtaining a sample with 60% or more women?
b. What proportion of all possible random samples of 80 faculty members that one could draw from the Penn State faculty would have between 30% and 40% women?
c. What is the probability of selecting a random sample of 150 Penn State faculty members and obtaining a sample with 45% or less of the sample being women?
d. What proportion of all possible samples of 50 Penn State faculty members would contain 55% or fewer women?

3. In a sentencing study of 150 prisoners convicted of the same crime, the mean sentence was found to be 132.3 months with a standard deviation of 34.7 months.
a. Find the 95% confidence interval for the sentence length for this offense.
b. Find the 90% confidence interval estimating the length of sentence for this offense.
c. Find the 50% confidence interval for length of this sentence.
d. Compare these three confidence intervals in regard to width and certainty that they include the population mean.

4. Suppose that a random sample of 25 people living in Mudville has a mean income of $25,000 with a standard deviation of $5,000.
a. What is your best single estimate (point estimate) of the income of all Mudville residents?
b. Construct a 95% confidence interval to estimate the mean income of all Mudville residents.
c. Suppose that the confidence interval established in (b) above is too wide. How could the width of the interval be reduced?

5. One fourth of 300 persons interviewed were opposed to a certain program.
Calculate an 80% confidence internal for the fraction of the population who are in opposition to the program..

6. Twelve cars traveling on an interstate highway are clocked at the following speeds (in miles per hour):
60 75 55 69 75 68
62 70 69 58 60 63
If the speed of cars traveling on this highway has a normal distribution, construct a 95% confidence interval for the mean speed of all cars traveling on this interstate.

7. In a sample of 400 seeds, 240 germinated.
a. Estimate the true germination rate with a 90% confidence interval,
b. Estimate the true germination rate with a 99% confidence interval.

8. Fifteen students randomly selected from a STAT 200 class kept a record of the number of hours they spent doing homework for the class during a given week. The following information was compiled:
Total number of hours spent by all 15 students combined (XX) = 90 Sum of the squares of the hours spent by the 15 students (XX2) = 554
Sum of the squares of the deviations of the individual student's times about the sample mean (£(X - X) =14
Assuming that, in the population, hours spent on doing STAT 200 homework is normally distributed, set up a 90% confidence interval estimating the mean number of hours per week all students in the STAT 200 class spent doing homework.

9. In a random sample of 200 Mudville residents interviewed in a recent survey, 130 reported that their community was "a good place to live."
a. What is your best estimate, based on these data, of the percentage of all Mudville residents who feel that their community is "a good place to live."
b. What is the "margin of error" of your estimate in (a) above?

10. Following are the red blood cell counts (in 106 cells per microliter) of a healthy person measured on each of 15 days.
5.4 5.2 5.0 5.2 5.5
5.3 5.4 5.2 5.1 5.3
5.3 4.9 5.4 5.2 5.2
Find a 95% confidence interval estimate of |j,, the true mean red blood cell count for this person, during the period of testing.

11. Suppose you want to estimate at a 99% confidence level, the amount of money people in Podunk spend each week for groceries. From an earlier study, you know that the standard deviation in such expenditures is $30. What is the minimum sample size you should use so that your estimate is within $5.00 of the population mean?

12. An experimenter has prepared a drug-dose level designed to induce sleep in 60% of all cases treated. How large a sample should be treated if he wants to estimate the true proportion within .04 with a probability of .95?

13. Suppose that you wish to estimate a population mean based on a random sample of n observations, and prior experience suggests that a = 12.7. If you wish to estimate n correct to within 1.6, with probability equal to .95, how many observations should be included in your sample?

14. Suppose you wish to estimate a binomial parameter p correct to within .04, with probability equal to .95. If you know that p is equal to some value between .1 and .3 and you want to be certain that your sample is large enough, how large should n be? (HINT: when calculating the standard error, use the value of p in the interval (.1 < p < .3) that will give the largest sample size.)

15. Suppose that you want to survey a random sample of registered voters in Pennsylvania to estimate what proportion will answer "yes" to a series of "yes/no" questions. You want to have a margin of error of no more than .02. (That is, you want to be 95% confident that your estimates are within two percentage points of the actual p-value.) How many people will you need to participate in your survey?