A principal at a certain school claims that the students in his school are above average intelligence. A random sample of thirty students IQ scores have a mean score of 112.5. Is there sufficient evidence to support the principal’s claim? The mean population IQ is 100 with a standard deviation of 15. Use 0.05 level of significance.

We have that

$n=30$

$\bar x=112$

$\mu = 100$

$\sigma=15$

$H_0: \mu=100$

$H_a:\mu>100$

The hypothesis test is right-tailed.

The population standard deviation is known and the sample size is large (n≥30) so we use z-test.

Let the significance level be 5% in this test, therefore Z0.05 = 1.64

The critical region is Z > 1.64

Test statistic:

$Z_{test}=\frac{\bar x -\mu}{\frac{\sigma}{\sqrt n}}=\frac{112 -100}{\frac{15}{\sqrt {30}}}=4.38$

Since 4.38 > 1.64 thus the Ztest falls in the rejection region we reject the null hypothesis.

At the 5% significance level the data do provide sufficient evidence to support the claim. We are 95% confident to conclude that the students in the school are above average intelligence.