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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.

    • x ¯ = size 12{ {overline {x}} ={}} {} ________
    • σ = size 12{σ} {} ________
    • s x = size 12{s rSub { size 8{x} } ={}} {} ________
    • n = size 12{n} {} ________
    • n 1 = size 12{n - 1} {} ________
  • Define the Random Variables X size 12{X} {} and X ¯ size 12{ {overline {X}} } {} , in words.
  • Which distribution should you use for this problem? Explain your choice.
  • Construct a 90% confidence interval for the population mean number of letters campers send home.
    • State the confidence interval.
    • Sketch the graph.
    • Calculate the error bound.
  • What will happen to the error bound and confidence interval if 500 campers are surveyed? Why?

    • 7.9
    • 2.5
    • 2.8
    • 20
    • 19
  • N ( 7 . 9, 2 . 5 20 ) size 12{N \( 7 "." 9, { { size 8{2 "." 5} } over { size 8{ sqrt {"20"} } } } \) } {}
    • CI: (6.98, 8.82)
    • EB: 0.92

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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.

  • Define the Random Variables X size 12{X} {} and P ' size 12{P'} {} , in words.
  • Which distribution should you use for this problem? Explain your choice.
  • Construct a 97% confidence interval for the population proportion of people over 50 who ran and died in the same eight–year period.
    • State the confidence interval.
    • Sketch the graph.
    • Calculate the error bound.
  • Explain what a “97% confidence interval” means for this study.

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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.

  • Which distribution should you use for this problem? Explain your choice.
  • Define the Random Variable X ¯ size 12{ {overline {X}} } {} , in words.
  • Construct a 95% confidence interval for the population mean cost of a used car.
    • State the confidence interval.
    • Sketch the graph.
    • Calculate the error bound.
  • Explain what a “95% confidence interval” means for this study.

  • t 83 size 12{t rSub { size 8{"83"} } } {}
  • mean cost of 84 used cars
    • CI: (5740.10, 7109.90)
    • EB = 684.90

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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.

  • Define the Random Variables X size 12{X} {} and P ' size 12{P'} {} , in words.
  • Which distribution should you use for this problem? Explain your choice.
  • Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem.
    • State the confidence interval.
    • Sketch the graph.
    • Calculate the error bound.
  • Suppose we want to lower the sampling error. What is one way to accomplish that?
  • The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ± 3%. In 1-3 complete sentences, explain what the ± 3% represents.

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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 size 12{X} {} and P ' size 12{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 worried a lot about the quality of education in our schools.
    • State the confidence interval.
    • Sketch the graph.
    • 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.

  • N ( 0.63 , (0.63)(0.37) 1000 )
    • CI: (0.60, 0.66)
    • EB = 0.03

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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.

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

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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?

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Try these multiple choice questions.

The next three problems refer to the following: According to a Field Poll, 79% of California adults (actual results are 400 out of 506 surveyed) feel that “education and our schools” is one of the top issues facing California. We wish to construct a 90% confidence interval for the true proportion of California adults who feel that education and the schools is one of the top issues facing California. (Source: http://field.com/fieldpollonline/subscribers/)

A point estimate for the true population proportion is:

  • 0.90
  • 1.27
  • 0.79
  • 400

C

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A 90% confidence interval for the population proportion is:

  • (0.761, 0.820)
  • (0.125, 0.188)
  • (0.755, 0.826)
  • (0.130, 0.183)

A

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The error bound is approximately

  • 1.581
  • 0.791
  • 0.059
  • 0.030

D

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The next two problems refer to the following:

A quality control specialist for a restaurant chain takes a random sample of size 12 to check the amount of soda served in the 16 oz. serving size. The sample mean is 13.30 with a sample standard deviation of 1.55. Assume the underlying population is normally distributed.

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

  • (12.42, 14.18)
  • (12.32, 14.29)
  • (12.50, 14.10)
  • Impossible to determine

B

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What is the error bound?

  • 0.87
  • 1.98
  • 0.99
  • 1.74

C

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

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

C

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The next two problems refer to the following:

Five hundred and eleven (511) homes in a certain southern California community are randomly surveyed to determine if they meet minimal earthquake preparedness recommendations. One hundred seventy-three (173) of the homes surveyed met the minimum recommendations for earthquake preparedness and 338 did not.

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.

  • (0.2975, 0.3796)
  • (0.6270, 6959)
  • (0.3041, 0.3730)
  • (0.6204, 0.7025)

C

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

  • 0.6614
  • 0.3386
  • 173
  • 338

A

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Questions & Answers

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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