# 7-4. Exercises

## 1. **BASIC**

**Ex 1.** Estimate the minimum sample size needed to form a confidence interval for the mean of a population having the standard deviation shown, meeting the criteria given.

1. $$σ = 30$$, 95% confidence, $$E = 10$$&#x20;
2. $$σ = 30$$, 99% confidence, $$E = 10$$&#x20;
3. $$σ = 30$$, 95% confidence, $$E = 5$$&#x20;

**Ex 2.** Estimate the minimum sample size needed to form a confidence interval for the mean of a population having the standard deviation shown, meeting the criteria given.

1. $$σ = 4$$, 95% confidence, $$E = 1$$&#x20;
2. $$σ = 4$$, 99% confidence, $$E = 1$$&#x20;
3. $$σ = 4$$, 95% confidence, $$E = 0.5$$&#x20;

**Ex 3.** Estimate the minimum sample size needed to form a confidence interval for the proportion of a population that has a particular characteristic, meeting the criteria given.

1. $$p ≈ 0.37$$ , 80% confidence, $$E = 0.05$$&#x20;
2. $$p ≈ 0.37$$, 90% confidence, $$E = 0.05$$&#x20;
3. $$p ≈ 0.37$$, 80% confidence, $$E = 0.01$$&#x20;

**Ex 4.** Estimate the minimum sample size needed to form a confidence interval for the proportion of a population that has a particular characteristic, meeting the criteria given.

1. $$p ≈ 0.81$$, 95% confidence, $$E = 0.02$$&#x20;
2. $$p ≈ 0.81$$, 99% confidence, $$E = 0.02$$&#x20;
3. $$p ≈ 0.81$$, 95% confidence, $$E = 0.01$$&#x20;

**Ex 5.** Estimate the minimum sample size needed to form a confidence interval for the proportion of a population that has a particular characteristic, meeting the criteria given.

1. 80% confidence, $$E = 0.05$$&#x20;
2. 90% confidence, $$E = 0.05$$&#x20;
3. 80% confidence, $$E = 0.01$$&#x20;

**Ex 6.** Estimate the minimum sample size needed to form a confidence interval for the proportion of a population that has a particular characteristic, meeting the criteria given.

1. 95% confidence, $$E = 0.02$$&#x20;
2. 99% confidence, $$E = 0.02$$&#x20;
3. 95% confidence, $$E = 0.01$$&#x20;

## **2. APPLICATIONS**

**Ex 7.** A software engineer wishes to estimate, to within 5 seconds, the mean time that a new application takes to start up, with 95% confidence. Estimate the minimum size sample required if the standard deviation of start up times for similar software is 12 seconds.

**Ex 8.** A real estate agent wishes to estimate, to within $2.50, the mean retail cost per square foot of newly built homes, with 80% confidence. He estimates the standard deviation of such costs at $5.00. Estimate the minimum size sample required.

**Ex 9.** An economist wishes to estimate, to within 2 minutes, the mean time that employed persons spend commuting each day, with 95% confidence. On the assumption that the standard deviation of commuting times is 8 minutes, estimate the minimum size sample required.

**Ex 10.** A motor club wishes to estimate, to within 1 cent, the mean price of 1 gallon of regular gasoline in a certain region, with 98% confidence. Historically the variability of prices is measured by $$σ=$0.03$$ . Estimate the minimum size sample required.

**Ex 11.** A bank wishes to estimate, to within $25, the mean average monthly balance in its checking accounts, with 99.8% confidence. Assuming $$σ=$250$$ , estimate the minimum size sample required.

**Ex 12.** A retailer wishes to estimate, to within 15 seconds, the mean duration of telephone orders taken at its call center, with 99.5% confidence. In the past the standard deviation of call length has been about 1.25 minutes. Estimate the minimum size sample required. (Be careful to express all the information in the same units.)

**Ex 13.** The administration at a college wishes to estimate, to within two percentage points, the proportion of all its entering freshmen who graduate within four years, with 90% confidence. Estimate the minimum size sample required.

**Ex 14.** A chain of automotive repair stores wishes to estimate, to within five percentage points, the proportion of all passenger vehicles in operation that are at least five years old, with 98% confidence. Estimate the minimum size sample required.

**Ex 15.** An internet service provider wishes to estimate, to within one percentage point, the current proportion of all email that is spam, with 99.9% confidence. Last year the proportion that was spam was 71%. Estimate the minimum size sample required.

**Ex 16.** An agronomist wishes to estimate, to within one percentage point, the proportion of a new variety of seed that will germinate when planted, with 95% confidence. A typical germination rate is 97%. Estimate the minimum size sample required.

**Ex 17.** A charitable organization wishes to estimate, to within half a percentage point, the proportion of all telephone solicitations to its donors that result in a gift, with 90% confidence. Estimate the minimum sample size required, using the information that in the past the response rate has been about 30%.

**Ex 18.** A government agency wishes to estimate the proportion of drivers aged 16–24 who have been involved in a traffic accident in the last year. It wishes to make the estimate to within one percentage point and at 90% confidence. Find the minimum sample size required, using the information that several years ago the proportion was 0.12.

## **3. ADDITIONAL EXERCISES**

**Ex 19.** An economist wishes to estimate, to within six months, the mean time between sales of existing homes, with 95% confidence. Estimate the minimum size sample required. In his experience virtually all houses are re-sold within 40 months, so using the Empirical Rule he will estimate *σ* by one-sixth the range, or $$40∕6=6.7$$ .

**Ex 20.** A wildlife manager wishes to estimate the mean length of fish in a large lake, to within one inch, with 80% confidence. Estimate the minimum size sample required. In his experience virtually no fish caught in the lake is over 23 inches long, so using the Empirical Rule he will estimate *σ* by one-sixth the range, or $$23∕6=3.8$$ .

**Ex 21.** You wish to estimate the current mean birth weight of all newborns in a certain region, to within 1 ounce (1/16 pound) and with 95% confidence. A sample will cost $400 plus $1.50 for every newborn weighed. You believe the standard deviations of weight to be no more than 1.25 pounds. You have $2,500 to spend on the study.

1. Can you afford the sample required?
2. If not, what are your options?

**Ex 22.** You wish to estimate a population proportion to within three percentage points, at 95% confidence. A sample will cost $500 plus 50 cents for every sample element measured. You have $1,000 to spend on the study.

1. Can you afford the sample required?
2. If not, what are your options?
