The school nurse thinks the average height of 7th graders has increased. The average height of a 7th grader ﬁve years ago was 145 cm with a standard deviation of 20 cm. She takes a random sample of 200 students and ﬁnds that the average height of her sample is 147 cm. Are 7th graders now taller than they were before? Conduct a single-tailed hypothesis test using a .05 signiﬁcance level to evaluate the null and alternative hypotheses. First, we develop our null and alternative hypotheses:
H0 :µ≤145 Ha :µ > 145
Choose α = .05. The critical value for this one tailed test is z=1.64. This is a one-tailed test, and a z-score of 1.64 cuts off 5% in the single tail. Any test statistic greater than 1.64 will be in the rejection region. Next, we calculate the test statistic for the sample of 7th graders.
z =(147−145) / 20√ 200
The calculated z−score of 1.414 is smaller than 1.64 and thus does not fall in the critical region. Our decision is to fail to reject the null hypothesis and conclude that the probability of obtaining a sample mean equal to 147 is likely to have been due to chance.