Homework

Announcements for 84 upcoming engineering conferences were randomly picked from a stack of IEEE Spectrum magazines. The mean length of the conferences was 3.94 days, with a standard deviation of 1.28 days. Assume the underlying population is normal.

1. In words, define the random variables X and $\overline{X}$.
2. Which distribution should you use for this problem? Explain your choice.
3. Construct a 95 percent confidence interval for the population mean length of engineering conferences.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.

A sample of 16 small bags of the same brand of candies was selected. Assume that the population distribution of bag weights is normal. The weight of each bag was then recorded. The mean weight was two ounces with a standard deviation of 0.12 ounces. The population standard deviation is known to be 0.1 ounce.

1. 1. $\overline{x}$ =________
   2. σ =________
   3. s_x =________
2. In words, define the random variable X.
3. In words, define the random variable $\overline{X}$.
4. Which distribution should you use for this problem? Explain your choice.
5. Construct a 90 percent confidence interval for the population mean weight of the candies.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
6. Construct a 98 percent confidence interval for the population mean weight of the candies.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
7. In complete sentences, explain why the confidence interval in Part f is larger than the confidence interval in Part e.
8. In complete sentences, give an interpretation of what the interval in Part f means.

What is meant by the term 90 percent confident when constructing a confidence interval for a mean?

1. If we took repeated samples, approximately 90 percent of the samples would produce the same confidence interval.
2. If we took repeated samples, approximately 90 percent of the confidence intervals calculated from those samples would contain the sample mean.
3. If we took repeated samples, approximately 90 percent of the confidence intervals calculated from those samples would contain the true value of the population mean.
4. If we took repeated samples, the sample mean would equal the population mean in approximately 90 percent of the samples.

The American Community Survey (ACS), part of the U.S. Census Bureau, conducts a yearly census similar to the one taken every 10 years, but with a smaller percentage of participants. The most recent survey estimates with 90 percent confidence that the mean household income in the United States falls between $69,720 and $69,922. Find the point estimate for mean U.S. household income and the error bound for mean U.S. household income.

In six packages of multicolored fruit snacks, there were five red snack pieces. The total number of snack pieces in the six bags was 68. We wish to calculate a 96 percent confidence interval for the population proportion of red snack pieces.

1. Define the random variables X and P′ in words.
2. Which distribution should you use for this problem? Explain your choice.
3. Calculate p′.
4. Construct a 96 percent confidence interval for the population proportion of red snack pieces per bag.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
5. Do you think that six packages of fruit snacks yield enough data to give accurate results? Why or why not?

Suppose that a committee is studying whether there is wasted time in our judicial system. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. The committee randomly surveyed 81 people who recently served as jurors. The sample mean wait time was 8 hr, with a sample standard deviation of 4 hr.

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 95 percent confidence interval for the population mean time wasted.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
5. Explain in a complete sentence what the confidence interval means.

Suppose that 14 children who were learning to ride two-wheel bikes were surveyed to determine how long they had to use training wheels. It was revealed that they used them an average of six months, with a sample standard deviation of three months. Assume that the underlying population distribution is normal.

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 $\overline{X}$ in words.
4. Which distribution should you use for this problem? Explain your choice.
5. Construct a 99 percent confidence interval for the population mean length of time using training wheels.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
6. Why would the error bound change if the confidence level were lowered to 90 percent?

A major business magazine published data on the best small firms in 2012. These were firms that have been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. Table 8.13 shows the ages of the corporate CEOs for a random sample of these firms.

Use the sample data to construct a 90 percent confidence interval for the mean age of CEOs for these top small firms. Use the Student's t-distribution.

In a recent sample of 84 used car sales costs, the sample mean was $6,425, with a standard deviation of $3,156. Assume the underlying distribution is approximately normal.

1. Which distribution should you use for this problem? Explain your choice.
2. Define the random variable $\overline{X}$ in words.
3. Construct a 95 percent confidence interval for the population mean cost of a used car.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
4. Explain what a 95 percent confidence interval means for this study.

A survey of the mean number of cents off given by coupons was conducted by randomly surveying one coupon per page from the coupons section of a local newspaper. The following data were collected: 20¢, 75¢, 50¢, 65¢, 30¢, 55¢, 40¢, 40¢, 30¢, 55¢, $1.50, 40¢, 65¢, 40¢. Assume the underlying distribution is approximately normal.

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 95 percent confidence interval for the population mean worth of coupons.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
5. If many random samples were collected with 14 samples as the size, which percentage of the confidence intervals constructed should contain the population mean worth of coupons? Explain why.

Which of the following is the error bound?

1. 0.87
2. 1.98
3. 0.99
4. 1.74

Suppose that the insurance companies did conduct a survey. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. We are interested in the population proportion of drivers who claim they always buckle up.

1. 1. x = __________
   2. n = __________
   3. p′ = __________
2. Define the random variables X and P′ in words.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 95 percent confidence interval for the population proportion who claim they always buckle up.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
5. If this survey were done by telephone, list three difficulties the companies might have in obtaining random results.

An article regarding dating and marriage recently appeared in a major newspaper. Of the 1,709 randomly selected adults, 315 identified themselves as ethnicity A, 323 identified themselves as ethnicity B, 254 identified themselves as ethnicity C, and 779 identified themselves as ethnicity D. In this survey, 86 percent of ethnicity B said that they would welcome a person of ethnicity A into their families. Among ethnicity C, 77 percent would welcome a person of ethnicity D into their families, 71 percent would welcome a person of ethnicity A, and 66 percent would welcome a person of ethnicity B.

1. We are interested in finding the 95 percent confidence interval for the percent of all ethnicity B adults who would welcome a person of ethnicity D into their families. 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 percent confidence interval.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.

Stanford University conducted a study of whether running is healthy for men and women over age 50. During the first eight years of the study, 1.5 percent of the 451 members of the 50-Plus Fitness Association died. We are interested in the proportion of people over 50 who ran and died in the same eight year period.

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 97 percent confidence interval for the population proportion of people over 50 who ran and died in the same 8-year period.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
4. Explain what a 97 percent confidence interval means for this study.

Refer to Exercise 8.123. Another question in the poll asked, “[How much are] you worried about the quality of education in our schools?” Sixty-three percent responded “a lot”. We are interested in the population proportion of adult Americans who are worried a lot about the quality of education in our schools.

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 percent confidence interval for the population proportion of adult Americans who are worried a lot about the quality of education in our schools.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
4. The sampling error given by the group of researchers who conducted the poll is ±3 percent. In one to three complete sentences, explain what the ±3 percent represents.

A 90 percent confidence interval for the population proportion is _______.

1. (0.761, 0.820)
2. (0.125, 0.188)
3. (0.755, 0.826)
4. (0.130, 0.183)

Find the confidence interval at the 90 percent confidence level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness.

1. (0.2975, 0.3796)
2. (0.6270, 0.6959)
3. (0.3041, 0.3730)
4. (0.6204, 0.7025)

On May 23, 2013, a polling group reported that of the 1,005 people surveyed, 76 percent of U.S. workers believe that they will continue working past retirement age. The confidence level for this study was reported at 95 percent with a ±3 percent margin of error.

1. Determine the estimated proportion from the sample.
2. Determine the sample size.
3. Identify CL and α.
4. Calculate the error bound based on the information provided.
5. Compare the error bound in Part d to the margin of error reported by the polling group. Explain any differences between the values.
6. Create a confidence interval for the results of this study.
7. A reporter is covering the release of this study for a local news station. How should she explain the confidence interval to her audience?

A polling group recently conducted a survey asking adults across the United States about music preferences. When asked, 80 of the 571 participants download music weekly.

1. Create a 99 percent confidence interval for the true proportion of American adults who download music weekly.
2. This survey was conducted through automated telephone interviews on May 6 and 7, 2013. The error bound of the survey compensates for sampling error, or natural variability among samples. List some factors that could affect the survey’s outcome that are not covered by the margin of error.
3. Without performing any calculations, describe how the confidence interval would change if the confidence level decreased from 99 percent to 90 percent.

In a recent poll, 9 of 48 respondents rated the likelihood of a certain event occurring in their community as likely or very likely. Use the plus-four method to create a 97 percent confidence interval for the proportion of American adults who believe that the event is likely or very likely. Explain what this confidence interval means in the context of the problem. A local poll in a New England town found that nine of 48 households think winter-proofing their cars is very important. Use the plus-four method to create a 97 percent confidence interval for the proportion of town residents who think winter-proofing their cars is very important. Explain what this confidence interval means in the context of this scenario.