Question

Use the Statistical Applet: ? ‑Value for a Test of One Proportion to answer the question.

You have taken a sample of ?=1000 individuals and asked them if they typically eat breakfast. Because eating breakfast is considered healthy, you would like to know if the proportion in the population who eat breakfast is more than two‑thirds, about 67% . When you asked the question of ?=20 people, you found ?=15 .

Which sample size will give more certain results? n = 1000.....n = 20

Answer 1

Let p be the proportion of people who eat breakfast.

We have to test, H0: p = 0.67 against H1: p > 0.67

The test-statistic is given by, Z = , where, = 0.67, p = 15/20 = 0.75, n = 20

Thus, Z = 0.8362.

Under H0, Z ~ N(0,1)

The p-value = P(Z > 0.8262) = 0.2043

If we assume 5% significance level, then we have, p-value > level of significance (0.05).

So, we fail to reject H0 and conclude that the proportion of people eating breakfast is not significantly more than 67%.

Sample size of 1000 will given more certain results, as increasing the sample size decreases the standard error of the estimates.