Past experience has shown that the scores of students who take a certain mathema
ID: 3157193 • Letter: P
Question
Past experience has shown that the scores of students who take a certain mathematics test are normally distributed with a mean of 75 and a variance of 36.
The mathematics department members would like to know whether this year's group of 16 students is typical. They decide to test the hypothesis that this year's students are typical versus the alternative that they are not typical. When the students take the test, the average score is 82. What conclusion should be drawn? Use a significance level of a=0.10(90% confidence interval).
Explanation / Answer
The hypotheses are
H0 : = 75 , this year's students are typical.
H1 : 75 , this year's students are atypical.
level of significance, = .10 . We are testing a hypothesis
about the true mean score of our group of students. It seems reasonable
to use X , the sample mean, as a test statistic. If the scores are each
distributed with mean 75 and variance 36, then the sample mean is
distributed normally with mean 75 and variance 36/n.
In this case n = 16. Thus Z = [(X 75) / (36/16)]
is distributed normally with mean 0 and variance 1 .
We wish numbers c and c that will determine a critical region, i.e.,
where z < c or z > c will lead to rejecting the null hypothesis.
Pr(z < c) + Pr(z > c) .10 . Equivalently Pr(z > c) = .05. Using the tables,
c = 1.65.
We now calculate z for this example,
Z = [(82 75) / (36/16)] = [7 / (2.25)] = [7 / (1.5)] = 4.67 .
Since 4.67 > 1.65, we reject H0 and conclude that this year's
students are atypical.