Question

A researcher in the social sciences is concerned that a sample of respondents to a mail-out survey may not be representative of the target population under study. Specifically, the researcher is concerned that the sample of respondents are significantly younger than the population average of 37. For the survey, the social scientist obtained a mean age of 22.4 years, and a standard deviation of 6.8 for a sample of 40 respondents. What can the researcher conclude if the probability of a Type I error is to be at most 0.01? (Show all Work)

Answer 1

Null and alternative hypotheses

Ho : = 37

Ha : < 37

Test statistic

t = (xbar - )/(s/√n)

t = (22.4 - 37)/(6.8/√40)

t = -13.58

tCritical for a = 0.01 and degrees of freedom = n - 1 = 39

t_{​​​Critical} = t_{​​​​​​a, n-1} = t_{​​​​​0.01 , 39}

t_{​​​​​​Critical} = -2.426

Decision rule : if t < tCritical we reject the null hypothesis otherwise we fail to reject the null hypothesis

Our t = -13.58 < -2.426

Decision : we reject the null hypothesis Ho

There is sufficient evidence to support the claim that respondents are significantly younger than the population average