In a statistics activity, students are asked to determine if there is a difference in the proportion of times that a spinning penny will land with tails up, and the proportion of times a spinning dime will land tails up. The students are instructed to spin the penny and the dime 30 times and record the number of times they land tails up. For one student, the penny lands tails side up 18 times, and the dime lands tails side up 20 times.

Based on the 98% confidence interval, (–0.36, 0.22), is there evidence of a difference in proportions of tails side up for a penny and a dime?

There is convincing evidence because the two sample proportions are different.

There is not convincing evidence because the interval contains 0.

There is not convincing evidence because the interval contains both negative and positive values for the true difference.

The 98% confidence interval (-0.36, 0.22) means we are 98% confident that the difference of proportions p1-p2 is somewhere between those two endpoints of -0.36 and 0.22 Values in this interval are possible values of p1-p2.

Note how 0 is in this interval. So it's possible that p1 - p2 = 0. Because of this, we fail to reject the null hypothesis. The null in this case is that p1 - p2 = 0 which can be restated as p1 = p2. The null is the assumption that the two population proportions (p1 and p2) are the same.

The alternative hypothesis is that p1-p2 is not zero. If we got a confidence interval like (0.22, 0.55), this example doesn't have 0 in the interval, then that would mean we reject the null since its unlikely that p1-p2 is equal to zero.

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In short,

If the confidence interval contains 0, then you fail to reject the null
If the confidence interval doesn't contain 0, then you reject the null.

In this case, we fail to reject the null and we don't have enough evidence to show that p1 = p2 is false. So for now, we assume it's true.