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. To estimate , we will use confidence intervals, as we did when estimating p. The basic form, as well as the theory, is the same: 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 point estimate for  ? What is the standard deviation for this estimator? What is the standard deviation for this estimator? How do we determine the appropriate number of standard deviations? How do we determine the appropriate number of standard deviations?

Confidence Intervals If  x has a normal distribution, then the confidence interval is If  x has a normal distribution, then the confidence interval isor If (  x –  )/(s/  n) has a t distribution, then the confidence interval is If (  x –  )/(s/  n) has a t distribution, then the confidence interval is

When to Use Z If If The population is normal (or nearly normal) and  is known, or The population is normal (or nearly normal) and  is known, or The population is not normal, but the sample size is at least 30, The population is not normal, but the sample size is at least 30, Then use Z. Then use Z.

When to Use t If If The population is normal (or nearly normal), and The population is normal (or nearly normal), and  is not known, and  is not known, and The sample size is less than 30, The sample size is less than 30, Then use t. 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. 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. Construct a 95% confidence interval for the true mean weight of such boxes.

Example Use Z. Why? Use Z. Why? n = 25. n = 25.  x =  x = Assume that  = Assume that  = Level of confidence = 95%, so z = Level of confidence = 95%, so z = 1.96.

Example The confidence interval is The confidence interval is 9.82  (1.96)(0.29/  25) = 9.82  = (9.706, 9.934).

TI-83 – Confidence Intervals When the standard normal distribution applies, do the following. When the standard normal distribution applies, do the following. Press STAT. Press STAT. Select TESTS. Select TESTS. Select ZInterval. Select ZInterval. A window appears requesting information. A window appears requesting information.

TI-83 – Confidence Intervals Select Data or Stats. Select Data or Stats. Assume we selected Stats. Assume we selected Stats. Enter . Enter . Enter  x. Enter  x. Enter n. Enter n. Enter the level of confidence. Enter the level of confidence. Select Calculate and press ENTER. Select Calculate and press ENTER.

TI-83 – Confidence Intervals A window appears containing A window appears containing The title “ZInterval”. The title “ZInterval”. The confidence interval in interval notation. The confidence interval in interval notation. The sample mean. The sample mean. The sample size. 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. 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. Construct a 99% confidence interval for the mean number of unoccupied seats.

Example Should we use Z or t? Why? Should we use Z or t? Why? n = 61. n = 61.  x =  x = s = 4.4. s = 4.4. Level of confidence = 99%. Find t. Level of confidence = 99%. Find t.

Example Consider again the t table (Table IV). Consider again the t table (Table IV). The degrees of freedom include every value up to 30, then jump to 40, 60, 120. The degrees of freedom include every value up to 30, then jump to 40, 60, 120. If the actual degrees of freedom are If the actual degrees of freedom are Between 30 and 40, use 30. Between 30 and 40, use 30. Between 40 and 60, use 40. Between 40 and 60, use 40. Between 60 and 120, use 60. Between 60 and 120, use 60. If they are beyond 120, use z. If they are beyond 120, use z.

Example The confidence interval is The confidence interval is 12.6  (2.660)(4.4/  61) = 12.6  = (11.101, ).

TI-83 – Confidence Intervals To use t, do the following. To use t, do the following. Press STAT. Press STAT. Select TESTS. Select TESTS. Select TInterval. Select TInterval. A window appears requesting information. A window appears requesting information.

TI-83 – Confidence Intervals Select Data or Stats. Select Data or Stats. Assume we selected Stats. Assume we selected Stats. Enter  x. Enter  x. Enter s. Enter s. Enter n. Enter n. Enter the level of confidence. Enter the level of confidence. Select Calculate and press ENTER. Select Calculate and press ENTER.

TI-83 – Confidence Intervals A window appears containing A window appears containing The title “TInterval”. The title “TInterval”. The confidence interval in interval notation. The confidence interval in interval notation. The sample mean. The sample mean. The sample standard deviation. The sample standard deviation. The sample size. The sample size.