Let's assume that over the years, a paper and pencil test of anxiety yields a mean score of 35 for all incoming college freshmen. We wish to determine whether the scores of a random sample of 20 new freshmen, with a mean of 30 and a standard deviation of 10, can be viewed as coming from this population. Test at the 0.05 level of significance.
Research statement: Does the population for all new freshmen differ from 35?
(1) State the null and alternative hypotheses in statistical notation.
Null hypothesis (H0): The population mean for all new freshmen is equal to 35.
Alternative hypothesis (Ha): The population mean for all new freshmen is not equal to 35.
(2) Report the z-critical value(s).
The z-critical value for a two-tailed test at the 0.05 level of significance is ±1.96.
(3) Compute the z-test statistic.
The z-test statistic can be calculated using the formula:
z = (sample mean - population mean) / (standard deviation / √sample size)
z = (30 - 35) / (10 / √20)
z = -5 / (10 / √20)
z = -5 / (10 / 4.472)
z = -5 / 2.236
z = -2.236
(4) State your statistical decision and explain why.
Since the calculated z-test statistic (-2.236) falls outside the range of the z-critical values (-1.96 to 1.96), we reject the null hypothesis. This means that there is sufficient evidence to suggest that the population mean for all new freshmen is different from 35.
(5) Summarize your conclusion.
Based on the results of the hypothesis test, we can conclude that the scores of the random sample of 20 new freshmen, with a mean of 30 and a standard deviation of 10, are significantly different from the population mean of 35.