Lower Tail Test of Population Proportion
The null hypothesis of the lower tail test about population proportion can be expressed as follows:
where p0 is a hypothesized lower bound of the true population proportion p.
Let us define the test statistic z in terms of the sample proportion and the sample size:
Suppose 60% of citizens voted in last election. 85 out of 148 people in a telephone survey said that they voted in current election. At 0.5 significance level, can we reject the null hypothesis that the proportion of voters in the population is above 60% this year?
The null hypothesis is that p ≥ 0.6. We begin with computing the test statistic.
> p0 = .6 # hypothesized value
> n = 148 # sample size
> z = (pbar−p0)/sqrt(p0∗(1−p0)/n)
> z # test statistic
We then compute the critical value at .05 significance level.
The test statistic -0.6376 is not less than the critical value of -1.6449. Hence, at .05 significance level, we do not reject the null hypothesis that the proportion of voters in the population is above 60% this year.
Instead of using the critical value, we apply the pnorm function to compute the lower tail p-value of the test statistic. As it turns out to be greater than the .05 significance level, we do not reject the null hypothesis that p ≥ 0.6.
We apply the prop.test function to compute the p-value directly. The Yates continuity correction is disabled for pedagogical reasons.
1−sample proportions test without continuity
data: 85 out of 148, null probability 0.6
X−squared = 0.4065, df = 1, p−value = 0.2619
alternative hypothesis: true p is less than 0.6
95 percent confidence interval: