Question

In 2010 polls indicated that 73% of Americans favored mandatory testing of students in public schools as a way to rate the school. This year in a poll of 1,000 Americans 71% favor mandatory testing for this purpose. Has public opinion changed since 2010?

We test the hypothesis that the percentage supporting mandatory testing is less than 73% this year. The p-value is 0.016.

Which of the following interpretation of this p-value is valid?

1. If 73% of Americans still favor mandatory testing this year, then there is a 1.6% chance that poll results will show 71% or fewer with this opinion.
2. The probability that Americans have changed their opinion on this issue since 2010 is 0.016.
3. There is a 1.6% chance that the null hypothesis is true.

Answer 1

Let X denote the number of Americans who favoured mandatory testing of students in public schools as a way to rate the school. In a sample of 1000 Americans, 71% i.e. 710 of them favor mandatory testing for this purpose.

Here X Bin( n,p) and X is sufficient for p.

We want to test the hypothesis, H​​​​​​0 : p = 0.73 v/s H​​​​​​a : p< 0.73 . The p-value is 0.016.

By the definition of p-value, the p-value or the probability value is the probability of obtaining the test results as extreme as the observed results of a statistical hypothesis test, assuming that the null hypothesis is true. The p-value gives the probability in favour of the alternative hypothesis.

The p-value is used as an alternative way to provide the smallest level of significance at which ​the null hypothesis would be rejected. That is, in this problem , there is 1.6% chance that the null hypothesis is false.

In the 2nd option, we have, the probability that the Americans have changed their opinion on this issue since 2010 is 0.016. There is a 1.6% chance that the null hypothesis is true. This contradicts the definition of p-value.

Therefore the 2nd option is not valid.

Now, under H​​​​​​0 , X Bin (n , 0.73) , n = 1000.

Hence under H​​​​​​0 , the distribution is completely specified. Therefore, X can be used as the test statistic.

Now for this test, p-value = P​​​​​​H0 ( X x​​​​​​0 ) where x​​​​​​0 is the observed value of X.

In this problem x​​​​​​0 = 710 (given) .

Therefore, p-value = P​​​​​​H0 ( X710) = P​​​​​​H0 (X/n 710/n) = P​​​​​​H0 ( p_hat 0.71) = 0.016 .

That is, if 73% Americans still favor mandatory testing this year ( assuming H​​​​​​0 is true) then there is 1.6 % that the poll results will show 71% or fewer with this opinion.

Therefore the 1st option is valid.