Question

Students at Hope College were tested to see if they could determine the difference between tap water and bottled water. Of the 63 students tested, 42 correctly identified which was which. We will assume the students tested are representative of all students at Hope College. Using the 2SD rule, we find that the 95% confidence interval is (0.541,0.793), and that the theory-based 95% confidence interval is (0.5503, 0.7831). Why are the 95% confidence intervals similar for the two methods? The intervals are similar because the validity conditions are not met. The intervals are similar because the sample size is large. Similar because validity conditions are met.

Answer 1

Sample proportion, p = 42/63 = 0.67

np(1-p) = 63 * 0.67 * (1 - 0.67) = 13.9293

Since np(1-p) > 10, the sample size is large enough to approximate the sampling distribution of proportion as normal distribution and apply empirical rule.

The sample can be assumed to be a random sample, the value of np(1-p) is greater than or equal to 10 and the sample size can be assumed to be less than or equal to 5% of the population size. Thus all validity conditions are met to apply empirical rule and 95% of the data values will fall within 2 standard deviations.

Similar, because validity conditions are met.