In another school a similar sample of student SAT math scores was taken. The principal of that school also believed that his students scored better than the national average. Suppose this principal collected a simple random sample of student SAT scores in math. The sample data collected had a mean student SAT score higher than 550 and the calculated P-value indicated that the null hypothesis should be rejected. In fact, the true population mean of that school's students' SAT scores is the same as the national mean score. What kind of error has been committed and why?a. Type I error because the principal rejected the null hypothesis when it was true.
b. Type I error because the principal failed to reject the null hypothesis when it was true.
c. Type II error because the principal rejected the null hypothesis when it was true.
d. Type II error because the principal failed to reject the null hypothesis when it was true.

We are given that principal of a school believed that his students scored better than the national average. For this principal collected a simple random sample of student SAT scores in math. The sample data collected had a mean student SAT score higher than 550 and the calculated P-value indicated that the null hypothesis should be rejected which means;

Null Hypothesis, [tex]H_0[/tex] : [tex]\mu[/tex] = 550 {means school's students' SAT scores is the same as the national mean score of 550}

Alternate Hypothesis, [tex]H_1[/tex] : [tex]\mu[/tex] > 550 {means school's students' SAT scores is higher than the national mean score of 550}

But, in fact, the true population mean of that school's students' SAT scores is the same as the national mean score and P-value indicates that the null hypothesis should be rejected.

Hence error has been occurred.

Type I error states that Probability of rejecting null hypothesis given the fact that [tex]H_0[/tex] is true and this is the case of our question as Principal had rejected the null hypothesis based on p-value but in actual [tex]H_0[/tex] was true.