Question

8. (15 pts) A person’s Intelligence Quotient (IQ) is determined via a series of test questions. The IQ score itself is designed to be approximately normally distributed with a mean value of 100 and a standard deviation of 15. In a sample of 20 students with behavioral problems, a school administrator observes an average IQ of 102.7. The administrator believes that students with behavioral issues have a different cognition from their peers, and thus a different average IQ. State the appropriate hypotheses; conduct a hypothesis test using α = 0.05 utilizing the classical approach, confidence interval approach, or p-value approach; state the decision regarding the hypotheses; and make a conclusion.

Answer 1

The provided sample mean is and the known population standard deviation is , and the sample size is

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation, will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is Zc​=1.96.

The rejection region for this two-tailed test is

(3) Test Statistics

The z-statistic is computed as follows:

(4) The decision about the null hypothesis

Since it is observed that ∣z∣=0.805≤zc​=1.96, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.4208, and since p=0.4208≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ is different than 100, at the 0.05 significance level.

Graphically

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