9.2 C ONFIDENCE I NTERVALS About the Population Mean when Population Standard Deviation is Unknown Obj: Use sample data to create a confidence interval without knowing population data

R EVIEW We created confidence intervals when μ was unknown and σ was known by using the formula as long as the sample size was 30 or above or we knew that the population was normally distributed. We no longer know what the value of σ is, so...

D ISTRIBUTION CHANGES... So we can no longer use the z-distribution, we have to use the t-distribution. Student’s t-Distribution is similar to the z-distribution. with n-1 degrees of freedom (Student is the pen name of William Sealey Gosset.)

T- DISTRIBUTION Properties The distribution changes depending on the number of degrees of freedom The distribution is centered at 0 and is symmetric about 0. The area under the curve is 1. As x increases or decreases without bound, the graph approaches, but never equals zero. The area in the tails is greater than in the tails of the standard normal distribution. As the sample size increases, the curve gets closer to the standard normal curve.

F INDING T - VALUES T-values are critical values written as t α such that α is the area under the t-distribution to the right of t α. Use Table V in the back of the book to find t-values. Find the t-value such that the area in the right tail is 0.05 with 25 degrees of freedom. Find the t-value such that the area in the right tail is 0.10 with 30 degrees of freedom.

C ONFIDENCE I NTERVALS To construct a confidence interval about μ with σ unknown, we use a formula very similar to the formula when σ is known: Or on the calculator: STAT; TESTS; TINTERVAL; choose either DATA or STATS; fill in missing data; CALCULATE

P RACTICE A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x, is found to be 50, and the sample deviation, s, is found to be 8. Construct a 98% confidence interval about μ if the sample size, n, is 20. Construct a confidence interval if n = 15. What effect did the sample size have on the margin of error?

P RACTICE A server at a restaurant wanted to estimate the mean tip percentage that she earns during dinner. She randomly selects 14 receipts from dinner, records the tip, and computes the tip rate Construct a 95% confidence interval for the tip percent. Assume a normal distribution. Assignment: page 4739 – 27 by 3