Question

1. A standardized test was administered to thousands of students with a mean score of 85 and a standard deviation of 8. A random sample or 50 students were given the same test and showed an average score of 83.20. Is there evidence to show that this group has a lower performance than the ones in general at 0.05 significance level?

Answer 1

Null hypothesis $H_0:\mu=85.$

Alternative hypothesis $H_a:\mu<85.$

Test statistic: $z=\frac{\bar x-\mu}{\frac{\sigma}{\sqrt{n}}}=\frac{83.20-85}{\frac{8}{\sqrt{50}}}=-1.59.$

P-value: $p=P(Z<-1.59)=0.0559.$

Since the P-value is greater than 0.05, fail to reject the null hypothesis.

There is no evidence to show that this group has a lower performance than the ones in general at 0.05 significance level.