Confidence Interval Estimation for a Population Mean Lecture 38 Section 10.3 Wed, Mar 30, 2005

(pt. est.)  (approp. no. of st. devs.) 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 Ifx has a normal distribution, then the confidence interval is or  If (x – )/(s/n) has a t distribution, then the confidence interval is

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

When to Use t If Then use t. The population is normal (or nearly normal), and  is not known, and The sample size is less than 30, Then use t.

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. 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 See Example 10.5, p. 591 – Cereal Boxes. Use Z. Why? n = 25. 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 . Enterx. 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.

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

Example See Example 10.6, p. 594 – Empty Seats Imply Dollars Lost. Should we use Z or t?

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. Enterx. 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.