Section 8.1 Estimating μ When σ is Known
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1 Chapter 8 Estimation Name Section 8.1 Estimating μ When σ is Known Objective: In this lesson you learned to explain the meanings of confidence level, error of estimate, and critical value; to find the critical value of a given confidence level; compute confidence intervals and sample size. Important Vocabulary Point Estimate Margin of Error Confidence Level Critical Value I. Estimating μ when σ is Known Assumptions About the Random Variable x Explain the meanings of confidence level, error of estimate, and critical value Find the critical value corresponding to a given confidence level Compute confidence intervals for μ when σ is known A point estimate The margin of Error What is a confidence level? For a confidence level c, the critical value z c 1
2 (Maximal) Margin of Error, E E = A c confidence interval for μ How to Find a Confidence Interval for μ When σ is Known Requirements Confidence interval for μ when σ is known What Does a Confidence Interval Tell Us? 2
3 II. Sample Size for Estimating the Mean μ How to Find the Sample Size n for Estimating μ when σ is Known Requirements Compute the sample size to be used for estimating a mean μ Formula for Sample Size 3
4 Section 8.1 Examples Estimating μ When σ is Known ( 1 ) Walter usually meets Julia at the track. He prefers to job 3 miles. From long experience, he knows that σ = 2.40 minutes for his jogging times. For a random sample of 90 jogging sessions, the mean time was x = minutes. Let μ be the mean jogging time for the entire distribution of Walter s 3-mile running times over the past several years. Find a 0.99 confidence interval for μ. a. Is the x distribution approximately normal? Do we know σ? b. What is the value of z 0.99? c. What is the value of E? d. What are the endpoints for a 0.99 confidence interval for μ? e. Explain what the confidence interval tells us. ( 2 ) A wildlife study is designed to find the mean weight of salmon caught by an Alaskan fishing company. A preliminary study of a random sample of 50 salmon showed s 2.15 pounds. How large a sample should be taken to be 99% confident that the sample mean x is within 0.20 pound of the true mean weight μ? 4
5 Section 8.2 Estimating μ When σ is Unknown Objective: In this lesson you learned about degrees of freedom and Student s t Distribution; how to compute confidence intervals for μ when σ is unknown. Important Vocabulary Student s t Distribution Degrees of Freedom (d. f. ) Critical Values I. Student s t Distribution Assume that x has a normal distribution with mean μ. Learn about degrees of freedom and Student s t Distribution Properties of a Student s t Distribution II. Using Table 4 to Find Critical Values for Confidence Intervals Convention for Using a Student s t Distribution Table Find critical values using degrees of freedom and confidence levels 5
6 III. Confidence Intervals for μ When σ is Unknown Margin of Error E = Compute confidence intervals for μ when σ is unknown How to Find a Confidence Interval for μ When σ is Unknown Requirements Confidence interval for μ when σ is unknown IV. Summary: Confidence Intervals for the Mean Situation I (Most Common) Situation II (almost never happens!) Which distribution should you use for x? 6
7 Section 8.2 Examples Estimating μ When σ is Unknown ( 1 ) Use Table 4 of the Appendix to find t c for a 0.90 confidence level for a t distribution with sample size n = 9. a. We find the column headed by c =. b. The degrees of freedom are given by d. f. = n 1 =. c. Read down the column found in part (a) until you reach the entry in the row headed by d. f. = 8. The value of t 0.90 is for a sample of size 9. d. Find t c for a 0.95 confidence level for a t distribution with sample size n = 9. ( 2 ) A company has a new process for manufacturing large artificial sapphires. In a trial run, 37 sapphires are produced. The distribution of weights is mound-shaped and symmetric. The mean weight for these 37 gems is x = 6.75 carats, and the sample standard deviation is s = 0.33 carat. Let μ be the mean weight for the distribution of all sapphires produced by the new process. a. Is it appropriate to use a Student s t distribution to compute a confidence interval for μ? b. What is d. f. for this setting? c. Use Table 4 of the Appendix to find t Note: d. f. = 36 is not in the table. Use the d. f. closest to 36 that is smaller than 36. d. Find E. e. Find a 95% confidence interval for μ. f. What does the confidence interval tell us in the context of the problem? 7
8 Section 8.3 Estimating p in the Binomial Distribution Objective: In this lesson you learned to compute margin of error for proportions, confidence intervals, sample size for estimating a proportion when there is (or is not) an estimate for p and how to interpret poll results. Important Vocabulary Point Estimate Margin of Error Confidence Interval for p I. Estimating p in the Binomial Distribution The point estimates for p and q are Compute the maximal margin of error for proportions using a given level of confidence Compute confidence intervals for p and interpret the results Margin of Error E = How to Find a Confidence Interval for a Proportion p Requirements Confidence interval for p 8
9 II. Interpreting Results From a Poll General Interpretation of Poll Results 1. Interpret poll results III. Sample Size for Estimating p How to Find the Sample Size n for Estimating a Proportion p Compute the sample size to be used for estimating a proportion p when we have an/no estimate for p 9
10 Section 8.3 Examples Estimating p in the Binomial Distribution ( 1 ) A random sample of 188 books purchased at a local bookstore showed that 66 of the books were murder mysteries. Let p represent the proportion of books sold by this store that are murder mysteries. a. What is a point estimate for p? b. Find a 90% confidence interval for p. c. What does the confidence interval you just computed mean in the context of this application? ( 2 ) A company is in the business of selling wholesale popcorn to grocery stores. The company buys directly from farmers. A buyer for the company is examining a large amount of corn from a certain farmer. Before the purchase is made, the buyer wants to estimate p, the probability that a kernel will pop. Suppose a random sample of n kernels is taken and r of these kernels pop. The buyer wants to be 95% sure that the point estimate p = r/n for p will be in error either way by less than a. If no preliminary study is made to estimate p, how large a sample should the buyer use? b. A preliminary study showed that p was approximately If the buyer uses the results of the preliminary study, how large a sample should he use? 10
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