Question

An IQ test scoring is designed to have mean of 100 and a standard deviation of 15. The scores are known to be normally distributed. A principal at a private school wants to investigate whether the average IQ of the students at her school is statistically different from the designed mean. She randomly samples from the school and has 15 students take the IQ test. The average test score of the sample is 95 and the sample standard deviation is 2. Use α = .10

1. Follow the 7 steps for hypothesis testing to solve this problem using the critical region method.

2. Calculate the p value.

3. Calculate the 90% confidence interval. Look to see if the hypothesized mean is in confidence intervals.

4. Do you come to the same conclusion using critical region, p value and confidence intervals?

Answer 1

1) H₀: = 100

    H₁: 100

The test statistic z = ()/()

                              = (95 - 100)/(15/)

                              = -1.29

At = 0.10, the critical values are +/- z_0.05 = +/- 1.645

Since the test statistic value is not less than the lower critical value (-1.29 > -1.645), so we should not reject the null hypothesis.

So at 0.1 significance level there is not sufficient evidence to conclude that the average IQ of the students is statistically different from the designed mean.
2) P-value = 2 * P(Z < -1.29)

                  = 2 * 0.0985

                  = 0.197

3) At 90% confidence interval the critical value is z_0.05 = 1.645

The 90% confidence interval for population mean is

+/- z_0.05 *

= 95 +/- 1.645 * 15/

= 95 +/- 6.371

= 88.629, 101.371

The hypothesized mean 100 lies in the above confidence interval .

4) Since the P-value > (0.197 > 0.10), so we should not reject the null hypothesis.

Since the confidence interval contains the hypothesized mean, so we should not reject the null hypothesis..

So we will get the same conclusion by using critical region, P-value and confidence intervals.