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8.2 Estimating Population Means

Learn how to estimate population means and proportions and compute associated margins of error and confidence intervals. Explore inferential statistics, confidence intervals, and determining sample sizes for accurate estimations.

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8.2 Estimating Population Means

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  1. 8.2 Estimating Population Means LEARNING GOAL Learn to estimate population means and compute the associated margins of error and confidence intervals.

  2. Review: Central Limit Theorem Suppose we take many random samples of size n for a variable with any distribution and record the means of each sample. Then: • The distribution of means will be approximately a normal distribution.

  3. Notation

  4. Inferential Statistics Inferential statistics uses sample statistics to estimate population parameters.

  5. Calories per day • Suppose a sample of 100 college students are asked to record the number of calories they eat in a day. The mean of the sample is 2560 calories with a standard deviation of 320 calories. • Does that indicate that all college students on average eat 2560 calories per day? • Would all samples of 100 students have a mean of 2560 calories? • 2560 calories is a single number estimate of the population mean, but we want to find an interval estimate or confidence interval.

  6. 95% Confidence Interval for Population Mean

  7. Back to calories per day • Suppose a sample of 100 college students are asked to record the number of calories they eat in a day. The mean of the sample is 2560 calories with a standard deviation of 320 calories. • Compute the 95% confidence interval for the number of calories consumed per day for all college students.

  8. What does a 95% Confidence Interval Mean? If we repeat the process of obtaining samples and constructing confidence intervals, in the long run 95% of the confidence intervals will contain the true population mean.

  9. Time to earn a bachelor’s degree • In a study of the length of time that students require to earn bachelor’s degrees, 80 students are randomly selected and they are found to have a mean of 4.8 years and a standard deviation of 2.2 years (based on data from the National Center for Education Statistics). • What is a single value estimate of the time it takes for the average student to complete a bachelor’s degree? • Find a 95% confidence interval for the time it takes for the average student to complete a bachelor’s degree.

  10. Sample Size Needed • Many times we know what margin of error we want, but need to know the sample size. • For a 95% confidence interval: • Use a past study, similar situation, or preliminary study to estimate .

  11. Sample Size for Weights of Quarters • The Tyco Video Game Corporation finds that it is losing income because of slugs used in its video games. The machines must be adjusted to accept coins only if they fall within set limits. In order to set those limits, the mean weight of quarters in circulation must be estimated. How many quarters must we randomly select if we want to be 95% confident that the sample mean is within 0.025 g of the true population mean for all quarters? Based on results from a sample of quarters we can estimate the standard deviation as 0.068 g.

  12. 8.3 Estimating Population Proportions LEARNING GOAL Learn to estimate population proportions and compute the associated margins of error and confidence intervals.

  13. Internet shopping • In a Gallup poll, 1025 randomly selected adults were surveyed and 297 of them said they used the Internet for shopping at least a few times a year. • What is a single number estimate for the proportion of adults who use the Internet for shopping? • Express the proportion as a percent.

  14. 95% Confidence Interval for a Population Proportion

  15. Internet shopping • In a Gallup poll, 1025 randomly selected adults were surveyed and 297 of them said they used the Internet for shopping at least a few times a year. • Find a 95% confidence interval for the proportion of all adults that use the Internet for shopping. • If a traditional retail store wants to estimate the percentage of adult Internet shoppers in order to determine the maximum impact of Internet shoppers on its sales, what percentage should be used?

  16. Survey Responses • In a survey of 1002 people, 701 said that they voted in a recent presidential election (based on data from ICR Research Group). Voting records show that 61% of eligible voters actually vote. • Find a 95% confidence interval estimate of the proportion of people who say that they voted. • Are the survey results consistent with the actual voter turnout of 61%? Why or why not?

  17. Choosing the Correct Sample Size To estimate a population proportion with a 95% degree of confidence, the sample size should be at least:

  18. Downloaded Songs • The music industry must adjust to the growing practice of consumers downloading songs instead of buying CDs. It therefore becomes important to estimate the proportion of songs that are currently downloaded. How many randomly selected individuals that have purchased music must be surveyed to determine the percentage that were obtained by downloading if we want to have 95% confidence with a margin of error of no more than one percentage point? Of no more than two percentage points?

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