Confidence Interval Estimation for a Population Mean Lecture 46 Section 10.3 Wed, Apr 14, 2004

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 If  x has a normal distribution, then the confidence interval is If  x has a normal distribution, then the confidence interval is  x  z  (  /  n) or  x  z  (s/  n). If (  x – u)/(s/  n) has a t distribution, then the confidence interval is If (  x – u)/(s/  n) has a t distribution, then the confidence interval is  x  t  (s/  n).

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 sample size is at least 30, 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.

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

Example See Example 10.5, p. 591 – Cereal Boxes. See Example 10.5, p. 591 – Cereal Boxes. 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).

Confidence Intervals on the TI-83 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.

Confidence Intervals on the TI-83 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.

Confidence Intervals on the TI-83 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.

Let’s Do It! Let’s do it! 10.6, p. 593 – How Much Beverage? Let’s do it! 10.6, p. 593 – How Much Beverage?

Example Under the right conditions, we must use the t distribution. Under the right conditions, we must use the t distribution. See Example 10.6, p. 594 – Empty Seats Imply Dollars Lost. See Example 10.6, p. 594 – Empty Seats Imply Dollars Lost.

Confidence Intervals on the TI-83 When the t distribution applies, do the following. When the t distribution applies, 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.

Confidence Intervals on the TI-83 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.

Confidence Intervals on the TI-83 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.

Assignment Page 598: Exercises 19, 21 – 28. Page 598: Exercises 19, 21 – 28. Page 606: Exercises 43 – 47, 49. Page 606: Exercises 43 – 47, 49.