9.2 CONFIDENCE INTERVALSAbout the Population Mean when Population Standard Deviation is Unknown

Obj: Use sample data to create a confidence interval without knowing population data

REVIEWWe 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 . . .

n

zxn

zx 2/2/ ,

DISTRIBUTION 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.)

nsxt

T-DISTRIBUTIONProperties 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.

FINDING T-VALUEST-values are critical values written as tα such

that α is the area under the t-distribution to the right of tα.

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.

CONFIDENCE INTERVALSTo 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

nstx

nstx 2/2/ ,

PRACTICEA 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?

PRACTICEA 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.

16.5 20.5 21.4 22.9 21.1 22.6 18.818.9 17.7 19.0 17.5 14.2 14.9 15.9

Construct a 95% confidence interval for the tip percent. Assume a normal distribution.

Assignment: page 473 9 – 27 by 3