The p-value is the probability of observing data at least as favorable to the alternative hy- pothesis as our current data set, if the null hypothesis were true. We typically use a summary statistic of the data, in this section the sample proportion, to help compute the p-value and evaluate the hypotheses.

EXAMPLE 5

Pew Research asked a random sample of 1000 American adults whether they supported the increased usage of coal to produce energy. Set up hypotheses to evaluate whether a majority of American adults support or oppose the increased usage of coal.

The uninteresting result is that there is no majority either way: half of Americans support and the other half oppose expanding the use of coal to produce energy. The alternative hypothesis would be that there is a majority support or oppose (though we do not known which one!) expanding the use of coal. If p represents the proportion supporting, then we can write the hypotheses as

H0: p = 0.5

HA: p 6= 0.5

In this case, the null value is p0 = 0.5.

When evaluating hypotheses for proportions using the p-value method, we will slightly modify how we check the success-failure condition and compute the standard error for the single proportion case. These changes aren’t dramatic, but pay close attention to how we use the null value, p0.