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CHAPTER SIX

Learn how to construct confidence intervals for the population mean using large and small samples, and explore the concepts of point estimate, interval estimate, level of confidence, and critical values.

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CHAPTER SIX

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  1. CHAPTER SIX Confidence Intervals

  2. Section 6.1 Confidence Intervals for the MEAN (Large Samples)

  3. Estimating Vocab • Point Estimate: a single value estimate for a population parameter. • Interval Estimate: a range of values used to estimate a population parameter. • Level of Confidence (c): the probability that the interval estimate contains the population parameter • The level of confidence, c, is the area under the curve between 2 z-scores called Critical Values

  4. Find the critical value zc necessary to construct a confidence interval at the given level of confidence. C = 0.85 C = 0.75

  5. More Vocab!

  6. Find the margin of error for the given values. • C = 0.90 s = 2.9 n = 50 • C = 0.975 s = 4.6 n = 100

  7. Confidence Intervals for the Population Mean

  8. Construct a C.I. for the Mean • 1. Find the sample mean and sample size. • 2. Specify σ if known. Otherwise, if n > 30 , find the sample standard deviation s. • 3. Find the critical value zc that corresponds with the given level of confidence. • 4. Find the margin of error, E. • 5. Find the left and right endpoints and form the confidence interval.

  9. Construct the indicated confidence interval for the population mean.

  10. 44. A random sample of 55 standard hotel rooms in the Philadelphia, PA area has a mean nightly cost of $154.17 and a standard deviation of $38.60. Construct a 99% confidence interval for the population mean. Interpret the results. • 46. Repeat Exercise 44, using a standard deviation of s = $42.50. Which confidence interval is wider? Explain.

  11. Sample Size: given c and E…

  12. Example from p 308 • 54. A soccer ball manufacturer wants to estimate the mean circumference of mini-soccer balls within 0.15 inch. Assume the population of circumferences is normally distributed. • A) Determine the minimum sample size required to construct a 90% C.I. for the population mean. Assume σ = 0.20 inch. • B) Repeat part (A) using σ = 0.10 inch. • C) Which standard deviation requires a larger sample size? Explain.

  13. Section 6.2 Confidence Intervals for the MEAN (Small Samples)

  14. The t – Distribution (table #5) • Used when the sample size n < 30 , the population is normally distributed, and σ is unknown. • t – Distribution is a family of curves. • Bell shaped, symmetric about the mean. • Total area under the t - curve is 1 • Mean, median, mode are equal to 0

  15. Uses Degrees of Freedom (d.f. =n–1) • d. f. are the # of free choices after a the sample mean is calculated. • To find the critical value, tc , use the t table. • Find the critical value, tc for c = 0.98, n = 20 • Find the critical value, tc for c = 0.95, n = 12

  16. Confidence Intervals and t - Distributions • 1. Find the sample mean, standard deviation, and sample size. • 2. ID the degrees of freedom, level of confidence and the critical value. • 3. Find the margin of error, E. • 4. Find the left and right endpoints and for the confidence interval.

  17. Construct the indicated C.I.

  18. Use a Normal or a t – Distribution to construct a 95% C.I. for the population mean. (from page 317) • 36. In a random sample of 13 people, the mean length of stay at a hospital was 6.2 days. Assume the population standard deviation is 1.7 days and the lengths of stay are normally distributed.

  19. Find the sample mean and standard deviation, the construct a 99% C.I. for the population mean. (from p 316) • 28. The weekly time spent (in hours) on homework for 18 randomly selected high school students: 12.0 11.3 13.5 11.7 12.0 13.0 15.5 10.8 12.5 12.3 14.0 9.5 8.8 10.0 12.8 15.0 11.8 13.0

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