Homework

A random survey of enrollment at 35 community colleges across the United States yielded the following figures: 6,414; 1,550; 2,109; 9,350; 21,828; 4,300; 5,944; 5,722; 2,825; 2,044; 5,481; 5,200; 5,853; 2,750; 10,012; 6,357; 27,000; 9,414; 7,681; 3,200; 17,500; 9,200; 7,380; 18,314; 6,557; 13,713; 17,768; 7,493; 2,771; 2,861; 1,263; 7,285; 28,165; 5,080; 11,622. Assume the underlying population is normal.

1. 1. $\stackrel{-}{x}$ = __________
   2. $s_x$ = __________
   3. n = __________
   4. n – 1 = __________
2. Define the random variables $X$ and $\stackrel{-}{X}$ in words.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 95% confidence interval for the population mean enrollment at community colleges in the United States.
   1. State the confidence interval.
   2. Sketch the graph.
5. What will happen to the error bound and confidence interval if 500 community colleges were surveyed? Why?

A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length of time they last is approximately normal. Researchers in a hospital used the drug on a random sample of nine patients. The effective period of the tranquilizer for each patient (in hours) was as follows: 2.7; 2.8; 3.0; 2.3; 2.3; 2.2; 2.8; 2.1; and 2.4.

1. 1. $\overline{x}$ = __________
   2. $s_x$ = __________
   3. n = __________
   4. n – 1 = __________
2. Define the random variable $X$ in words.
3. Define the random variable $\stackrel{-}{X}$ in words.
4. Which distribution should you use for this problem? Explain your choice.
5. Construct a 95% confidence interval for the population mean length of time.
   1. State the confidence interval.
   2. Sketch the graph.
6. What does it mean to be “95% confident” in this problem?

The Federal Election Commission (FEC) collects information about campaign contributions and disbursements for candidates and political committees each election cycle. A political action committee (PAC) is a committee formed to raise money for candidates and campaigns. A Leadership PAC is a PAC formed by a federal politician (senator or representative) to raise money to help other candidates’ campaigns.

The FEC has reported financial information for 556 Leadership PACs that operating during the 2011–2012 election cycle. The following table shows the total receipts during this cycle for a random selection of 30 Leadership PACs.

$\overline{x}=\$251,854.23$

$s=\$521,130.41$

Use this sample data to construct a 95% confidence interval for the mean amount of money raised by all Leadership PACs during the 2011–2012 election cycle. Use the Student's t-distribution.

Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. To accomplish this, the records of 225 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats.

1. 1. $\overline{x}$ = __________
   2. $s_x$ = __________
   3. n = __________
   4. n-1 = __________
2. Define the random variables $X$ and $\overline{X}$ in words.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 92% confidence interval for the population mean number of unoccupied seats per flight.
   1. State the confidence interval.
   2. Sketch the graph.

Six different national brands of chocolate chip cookies were randomly selected at the supermarket. The grams of fat per serving are as follows: 8; 8; 10; 7; 9; 9. Assume the underlying distribution is approximately normal.

1. Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets.
   1. State the confidence interval.
   2. Sketch the graph.
2. If you wanted a smaller error bound while keeping the same level of confidence, what should have been changed in the study before it was done?
3. Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies.
4. Calculate the mean.
5. Is the mean within the interval you calculated in part a? Did you expect it to be? Why or why not?

Find the 95% Confidence Interval for the true population mean for the amount of soda served.

1. (12.42, 14.18)
2. (12.32, 14.29)
3. (12.50, 14.10)
4. Impossible to determine

Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car.

1. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03?
2. If it were later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?

According to a recent survey of 1,200 people, 61% feel that the president is doing an acceptable job. We are interested in the population proportion of people who feel the president is doing an acceptable job.

1. Define the random variables X and P′ in words.
2. Which distribution should you use for this problem? Explain your choice.
3. Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job.
   1. State the confidence interval.
   2. Sketch the graph.

Refer to the information in Exercise 8.117.

1. Construct three 95% confidence intervals.
   1. percent of all Asians who would welcome a White person into their families.
   2. percent of all Asians who would welcome a Latino into their families.
   3. percent of all Asians who would welcome a Black person into their families.
2. Even though the three point estimates are different, do any of the confidence intervals overlap? Which?
3. For any intervals that do overlap, in words, what does this imply about the significance of the differences in the true proportions?
4. For any intervals that do not overlap, in words, what does this imply about the significance of the differences in the true proportions?

A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was “What is the main problem facing the country?” Twenty percent answered “crime.” We are interested in the population proportion of adult Americans who feel that crime is the main problem.

1. Define the random variables X and P′ in words.
2. Which distribution should you use for this problem? Explain your choice.
3. Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem.
   1. State the confidence interval.
   2. Sketch the graph.
4. Suppose we want to lower the sampling error. What is one way to accomplish that?
5. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ±3%. In one to three complete sentences, explain what the ±3% represents.

A point estimate for the true population proportion is:

1. 0.90
2. 1.27
3. 0.79
4. 400

The point estimate for the population proportion of homes that do not meet the minimum recommendations for earthquake preparedness is ______.

1. 0.6614
2. 0.3386
3. 173
4. 338

A national survey of 1,000 adults was conducted on May 13, 2013 by Rasmussen Reports. It concluded with 95% confidence that 49% to 55% of Americans believe that big-time college sports programs corrupt the process of higher education.

1. Find the point estimate and the error bound for this confidence interval.
2. Can we (with 95% confidence) conclude that more than half of all American adults believe this?
3. Use the point estimate from part a and n = 1,000 to calculate a 75% confidence interval for the proportion of American adults that believe that major college sports programs corrupt higher education.
4. Can we (with 75% confidence) conclude that at least half of all American adults believe this?

Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a 95% confidence interval for the mean height of male Swedes. Forty-eight male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches.

1. 1. $\stackrel{-}{x}$ =________
   2. σ =________
   3. n =________
2. In words, define the random variables X and $\stackrel{-}{X}$.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 95% confidence interval for the population mean height of male Swedes.
   1. State the confidence interval.
   2. Sketch the graph.
5. What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? Why?

Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. It randomly surveys 100 people. The sample mean is 23.6 hours. There is a known standard deviation of 7.0 hours. The population distribution is assumed to be normal.

1. 1. $\stackrel{-}{x}$ =________
   2. σ =________
   3. n =________
2. In words, define the random variables X and $\stackrel{-}{X}$.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 90% confidence interval for the population mean time to complete the tax forms.
   1. State the confidence interval.
   2. Sketch the graph.
5. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?
6. If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Why?
7. Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour. How would the number of people the firm surveys change? Why?

A camp director is interested in the mean number of letters each child sends during his or her camp session. The population standard deviation is known to be 2.5. A survey of 20 campers is taken. The mean from the sample is 7.9 with a sample standard deviation of 2.8.

1. 1. $\stackrel{-}{x}$ =________
   2. σ =________
   3. n =________
2. Define the random variables X and $\stackrel{-}{X}$ in words.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 90% confidence interval for the population mean number of letters campers send home.
   1. State the confidence interval.
   2. Sketch the graph.
5. What will happen to the error bound and confidence interval if 500 campers are surveyed? Why?

The Federal Election Commission collects information about campaign contributions and disbursements for candidates and political committees each election cycle. During the 2012 campaign season, there were 1,619 candidates for the House of Representatives across the United States who received contributions from individuals. Table 8.5 shows the total receipts from individuals for a random selection of 40 House candidates rounded to the nearest $100. The standard deviation for this data to the nearest hundred is σ = $909,200.

1. Find the point estimate for the population mean.
2. Using 95% confidence, calculate the error bound.
3. Create a 95% confidence interval for the mean total individual contributions.
4. Interpret the confidence interval in the context of the problem.

If the confidence interval is change to a higher probability, would this cause a lower, or a higher, minimum sample size?

If the tolerance is reduced by half, how would this affect the minimum sample size?

If the value of p is reduced, would this necessarily reduce the sample size needed?

Is it acceptable to use a higher sample size than the one calculated by $\frac{z^{2}pq}{e^2}$?

A company has been running an assembly line with 97.42%% of the products made being acceptable. Then, a critical piece broke down. After the repairs the decision was made to see if the number of defective products made was still close enough to the long standing production quality. Samples of 500 pieces were selected at random, and the defective rate was found to be 0.025%.

1. Is this sample size adequate to claim the company is checking within the 90% confidence interval?
2. The 95% confidence interval?