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9.2 Confidence Intervals for Population Means (σ unknown) Confidence intervals: We wish to use the sample mean as a point estimate of the population mean. The best we can do is construct an interval which contains the mean, with a certain ‘level of confidence’. __________________________________________________________________ Recall that sample means are normally distributed if the population is normal or if n ≥ 30. __________________________________________________________________ Suppose we know σ. We can calculate 𝜎𝑥̅ . We also know 95% of all sample means lie within 1.96 s.d. of μ. [use invNorm(.975) ≈ 1.96] i.e. – μ – 1.96𝜎𝑥̅ < 𝑥̅ < μ + 1.96𝜎𝑥̅ (95% chance of being true) or 𝑥̅ – 1.96𝜎𝑥̅ < 𝜇 < 𝑥̅ + 1.96𝜎𝑥̅ which gives us a “95% confidence interval” of (𝑥̅ – 1.96𝜎𝑥̅ , 𝑥̅ + 1.96𝜎𝑥̅ ). __________________________________________________________________ However, if σ is not known, we will use t-scores: 𝑡= 𝑥− 𝜇 𝑠 √𝑛 __________________________________________________________________ These scores are not normally distributed. Their distribution is called the Student’s t-distribution (with n-1 degrees of freedom). Properties of the t-distribution 1. Distribution varies as n varies. 2. Bell-shaped and symmetric about 0. 3. Area under the curve is 1. 4. Graph approaches zero in both tails. 5. Area in tails is slightly larger than for normal distribution. A (1-α)100% confidence interval will be given by: lower bound: 𝑥̅ − 𝑡𝛼 ∙ 2 𝑠 √𝑛 upper bound: 𝑥̅ + 𝑡𝛼 ∙ 2 𝑠 √𝑛 __________________________________________________________________ 𝑡𝛼 ∙ 2 𝑠 √𝑛 is called the ‘margin of error’ (half the length of the confidence interval) __________________________________________________________________ Example: (3rd ed.) A random sample of size n is drawn from a population that is normally distributed. The sample mean is found to be 108 and the sample standard deviation is found to be 10. a. Construct a 96% confidence interval for μ if the sample size is 25. b. Construct a 96% confidence interval for μ if the sample size is 10. c. Construct a 90% confidence interval for μ if the sample size is 25. Compare the above answers, noting relationships of n, α and the margin of error. d. Could we have computed the above confidence intervals if the population was not normally distributed? __________________________________________________________________ Use Stats/Tests/TInterval __________________________________________________________________ Example: (p.450, #36ac) [Note text answer.] __________________________________________________________________ Practice: (3rd ed.) A random sample has a mean of 18.4 and a standard deviation of 4.5. a. Construct a 95% confidence interval for μ if the sample size is 35. b. Construct a 95% confidence interval for μ if the sample size is 50. c. Construct a 99% confidence interval for μ if the sample size is 35. d. If the sample size was 15, what conditions must be satisfied to compute the confidence interval? __________________________________________________________________ Determining Sample Size 𝑧𝛼⁄2 ∙𝑠 2 𝑛=( 𝐸 ) [round up]