CK-12 Probability and Statistics - Advanced

7.6. Student’s t-Distribution http://www.ck12.org

z=

x ̄−μ 0

σ/

√

n

where ̄xis the sample mean,μ 0 is the hypothesized mean stated in the null hypothesisH 0 :μ=μ 0 σ, is the population
standard deviation andnis the sample size.

However the above formula cannot be used to determine how far a sample mean is from the hypothesized mean
because the standard deviation of the population is not known. If the value ofσis unknown,sis substituted forσ
andtforz. Thetstands for the “test statistic,” and it is given by the formula:

t=

x ̄−μ 0

s/

√

n

where ̄xis the sample meanμ 0 is the population mean,sis the standard deviation of the sample andnis the sample
size. The population meanμis unknown but an estimate for this value is used. Thet-test will be used to determine
the difference between the sample mean and the hypothesized mean. The null hypothesis that is being tested is
H 0 :μ=μ 0

So, suppose you want to see if a hypothesized mean passes a 95% level of confidence. The corresponding confidence
interval can be determined by using the graphing calculator:

Press ENTER

x=the number of successes in the sample and

n=the sample size

Press ENTER again. The confidence level will appear on the next screen. The value fortcan now be compared with
this interval to tell us whether or not the hypothesized mean can be accepted or rejected for this level of confidence.

Example:

The masses of newly produced bus tokens are estimated to have a mean of 3.16 grams. A random sample of 11
tokens was removed from the production line and the mean weight of the tokens was calculated as 3.21 grams with
a standard deviation of 0.067. What is the value of the test statistic for a test to determine how the mean differs from
the estimated mean?

Solution: