8.1 Homework  (Page 5/5)

A camp director is interested in the mean number of letters each child sends during his/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.

• a
  • i $\overline{x}=\text{\_\_\_\_\_\_\_\_}$
  • ii $\sigma =\text{\_\_\_\_\_\_\_\_}$
  • iii $s_x=\text{\_\_\_\_\_\_\_\_}$
  • iv $n=\text{\_\_\_\_\_\_\_\_}$
  • v $n-1=\text{\_\_\_\_\_\_\_\_}$
• b Define the Random Variables $X$ and $\overline{X}$ , in words.
• c Which distribution should you use for this problem? Explain your choice.
• d Construct a 90% confidence interval for the population mean number of letters campers send home.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the error bound.
• e What will happen to the error bound and confidence interval if 500 campers are surveyed? Why?

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% 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.

• a Define the Random Variables $X$ and $P\text{'}$ , in words.
• b Which distribution should you use for this problem? Explain your choice.
• c Construct a 97% confidence interval for the population proportion of people over 50 who ran and died in the same eight–year period.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the error bound.
• d Explain what a “97% confidence interval” means for this study.

In a recent sample of 84 used cars sales costs, the sample mean was $6425 with a standard deviation of $3156. Assume the underlying distribution is approximately normal.

• a Which distribution should you use for this problem? Explain your choice.
• b Define the Random Variable $\overline{X}$ , in words.
• c Construct a 95% confidence interval for the population mean cost of a used car.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the error bound.
• d Explain what a “95% confidence interval” means for this study.

A telephone poll of 1000 adult Americans was reported in an issue of Time Magazine . One of the questions asked was “What is the main problem facing the country?” 20% answered “crime”. We are interested in the population proportion of adult Americans who feel that crime is the main problem.

• a Define the Random Variables $X$ and $P\text{'}$ , in words.
• b Which distribution should you use for this problem? Explain your choice.
• c Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the error bound.
• d Suppose we want to lower the sampling error. What is one way to accomplish that?
• e The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ± 3%. In 1-3 complete sentences, explain what the ± 3% represents.

Refer to the above problem. Another question in the poll was “[How much are] you worried about the quality of education in our schools?” 63% 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 $P\text{'}$ , 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 worried a lot about the quality of education in our schools.
   • i State the confidence interval.
   • ii Sketch the graph.
   • iii Calculate the error bound.
4. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ± 3%. In 1-3 complete sentences, explain what the ± 3% represents.

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.

• a Calculate a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the error bound.
• b 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?
• c Go to the store and record the grams of fat per serving of six brands of chocolate chip cookies.
• d Calculate the mean.
• e Is the mean within the interval you calculated in part (a)? Did you expect it to be? Why or why not?

A confidence interval for a proportion is given to be (– 0.22, 0.34). Why doesn’t the lower limit of the confidence interval make practical sense? How should it be changed? Why?

A point estimate for the true population proportion is:

• A 0.90
• B 1.27
• C 0.79
• D 400

A 90% confidence interval for the population proportion is:

• A (0.761, 0.820)
• B (0.125, 0.188)
• C (0.755, 0.826)
• D (0.130, 0.183)

The error bound is approximately

• A 1.581
• B 0.791
• C 0.059
• D 0.030

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

• A (12.42, 14.18)
• B (12.32, 14.29)
• C (12.50, 14.10)
• D Impossible to determine

What is the error bound?

• A 0.87
• B 1.98
• C 0.99
• D 1.74

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

• A If we took repeated samples, approximately 90% of the samples would produce the same confidence interval.
• B If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean.
• C If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.
• D If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples.

Find the Confidence Interval at the 90% Confidence Level for the true population proportion of southern California community homes meeting at least the minimum recommendations for earthquake preparedness.

• A (0.2975, 0.3796)
• B (0.6270, 6959)
• C (0.3041, 0.3730)
• D (0.6204, 0.7025)

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

• A 0.6614
• B 0.3386
• C 173
• D 338