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Confidence Interval Estimation for a Population Mean Lecture 33 Section 10.3 Tue, Nov 14, 2006 Confidence Intervals To estimate , we will use confidence intervals, as we did when estimating p. The basic form, as well as the theory, is the same: (pt. est.) (approp. no. of st. devs.) Confidence Intervals What is the point estimate for ? What is the standard deviation for this estimator? How do we determine the appropriate number of standard deviations? Confidence Intervals Ifx has a normal distribution, then the confidence interval is or x z / n x z s / n If (x – )/(s/n) has a t distribution, then the confidence interval is x t s/ n When to Use Z If The population is normal (or nearly normal) and is known, or The population is not normal, but the sample size is at least 30, Then use Z. When to Use t If The population is normal (or nearly normal), and is not known, and The sample size is less than 30, Then use t. Example Example 10.4, p. 641: The Kellogg Corporation controls approximately a 43% share of the ready-to-eat cereal market worldwide. A popular cereal is Corn Flakes. Suppose the weights of full boxes of a certain kind of cereal are normally distributed with a population standard deviation of 0.29 ounces. A random sample of 25 boxes produced a mean weight of 9.82 ounces. Construct a 95% confidence interval for the true mean weight of such boxes. Example Use Z. Why? n = 25. x = 9.82. Assume that = 0.29. Level of confidence = 95%, so z = 1.96. Example The confidence interval is 9.82 (1.96)(0.29/25) = 9.82 0.114 = (9.706, 9.934). TI-83 – Confidence Intervals When the standard normal distribution applies, do the following. Press STAT. Select TESTS. Select ZInterval. A window appears requesting information. TI-83 – Confidence Intervals Select Data or Stats. Assume we selected Stats. Enter . Enterx. Enter n. Enter the level of confidence. Select Calculate and press ENTER. TI-83 – Confidence Intervals A window appears containing The title “ZInterval”. The confidence interval in interval notation. The sample mean. The sample size. Example Example 10.5, p. 643: Unoccupied seats on flights cause airlines to lose revenue. Suppose that a large airline wants to estimate its average number of unoccupied seats per flight from Detroit to Minneapolis over the past month. To accomplish this, the records of 61 such flights were randomly selected, and the number of unoccupied seats was recorded for each of the sampled flights. The sample mean is 12.6 and sample standard deviation is 4.4 seats. Construct a 99% confidence interval for the mean number of unoccupied seats. Example Should we use Z or t? Why? n = 61. x = 12.6. s = 4.4. Level of confidence = 99%. Find t. Example Consider again the t table (Table IV). The degrees of freedom include every value up to 30, then jump to 40, 60, 120. If the actual degrees of freedom are Between 30 and 40, use 30. Between 40 and 60, use 40. Between 60 and 120, use 60. If they are beyond 120, use z. Example The confidence interval is 12.6 (2.660)(4.4/61) = 12.6 1.499 = (11.101, 14.099). TI-83 – Confidence Intervals To use t, do the following. Press STAT. Select TESTS. Select TInterval. A window appears requesting information. TI-83 – Confidence Intervals Select Data or Stats. Assume we selected Stats. Enterx. Enter s. Enter n. Enter the level of confidence. Select Calculate and press ENTER. TI-83 – Confidence Intervals A window appears containing The title “TInterval”. The confidence interval in interval notation. The sample mean. The sample standard deviation. The sample size.