Question

A sports psychologist is part of a team of researchers collecting descriptive psychological, mental, performance and physiological data on male and female high school athletes. One of the variables is Intelligence Quotient (IQ) as assessed by the Stanford-Binet Intelligence Scale (5^th Ed. 2003). A sample of athletes (n=61) provided the following statistics: mean±s_X = 97±16. The parameter µ for IQ is thought to be 100.   Test H₀: X=µ at α=0.05÷2 (a 2-tailed test).

Step #1 – State H0: and HA:

Step #2 – State the criterion for attaining statistical significance

Step #3- Correct statistical procedure is a one-sample t-test

Step #4 – Find the critical 2-tailed statistical value

Step #5 – Draw the picture

Step #6 Calculate the statistical value t=mean-μsXn

Step #7 – Statistical Decision

Step #8 - Conclusion

Answer 1

The provided sample mean is 97 and the sample standard deviation is s = 16, and the sample size is n = 61

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ = 100

The Intelligence Quotient (IQ) of athletes is 100

Ha: μ ≠ 100

The Intelligence Quotient (IQ) of athletes is not 100

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown 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 t_c = 2

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that |t| = 1.464 < t_c = 2, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.1483 , and since p = 0.1483 > 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 Intelligence Quotient (IQ) of athletes is different than 100, at the 0.05 significance level.