example: A study was done to determine if 12- to 15-year-old girls who want to be engineers
differ in IQ from the average of all girls. The mean IQ of all girls in this age range is known to
be about 100 with a standard deviation of 15. A random sample of 49 girls who state that they
want to be engineers is selected and their IQ is measured. The mean IQ of the girls in the
sample is 104.5. Does this finding provide evidence, at the 0.05 level of significance, that the
mean IQ of 12- to 15-year-old girls who want to be engineers differs from the average?
Assume that the population standard deviation is 15 (σ = 15).
solution 1 (test statistic approach): The solution to this problem will be put into a form that
emphasizes the format required when writing out solutions on the AP exam.
I . Let μ = the true mean IQ of girls who want to be engineers.
H 0 : μ = 100.H (^) A : μ ≠ 100.
(The alternative is two-sided because the problem wants to know if the mean IQ “differs”
from 100. It would have been one-sided if it had asked whether the mean IQ of girls who
want to be engineers is higher than average.)
II . Since σ is known, we will use a one-sample z -test at α = 0.05.
Conditions:
• The problem states that we have a random sample.
• Sample size is large.
• s is known.
III . P -value = 2(1–0.9821) = 0.0358 (from Table A).
The TI-83/84 gives 2 × normalcdf(2.10,100)=0.0357.
IV . Because P < α, we reject H 0 . We have strong evidence that the true mean IQ for girls who