## Hypothesis Test for a Population Proportion

Our main goal is in finding the probability of a difference between a sample mean *p̂* and the claimed value of the population proportion, *p*_{0}.

In order for the sampling distribution of a sample proportion *p̂* to be approximately normal with mean *μ* = *p̂* and standard deviation

the following 3 conditions need to be met:

- The sample was obtained through a simple random sample process.
*n*⋅*p*⋅ (1 –*p*) ≥ 10*n*≤ 0.05 ⋅*N*, where*n*is the sample size and*N*is the size of the population.

**Example**

In 1995, 40% of adults aged 18 years or older reported that they had “a great deal” of confidence in the public schools. On June 1, 2005, the Gallup Organization (www.gallup.com) released results of a poll in which 372 of 1004 adults aged 18 years or older stated that they had “a great deal” of confidence in public schools. Does the evidence suggest at the *α* = 0.05 significance level that the proportion of adults aged 18 years or older having “a great deal” of confidence in the public schools is significantly lower in 2005 than the 1995 proportion?

**Step 1**: State the null and alternative hypotheses.

Basically, the goal of this problem is to see whether attitudes about public schooling have changed over time. We are asked to use the results from 1995 as the “baseline” and see whether, ten years later, attitudes are *lower*. Thus:

*H*_{0}:*p*= 0.40*H*_{α}:*p*< 0.40

Notice that this is a one-tail test since the question in the example wants to know whether confidence levels are LOWER.

**Step 2**: Determine the level of significance.

We are asked to use α = 0.05.

**Step 3**: Calculate the test statistic.

We first need to identify the sample proportion and standard deviation from the information given in the problem. We see that:

Using this information, the value of the test statistic is:

So our sample proportion is just under 2 standard deviations below the claimed value of the population proportion.

**Step 4**: Determine the *P*-value and the level of significance.

Using a table of standard normal values with a *z*-value of *z*_{0} = -1.91 we find that the probability value is 0.0281. Using technology (which doesn’t do as much rounding as we do with our calculations), we find that the probability value is 0.0282691712.

The question provided us with a significance level of 5%. Thus, *α* = 0.05 .

**Step 5**: Make appropriate conclusions

Comparing our *P*-value with the level of significance, one can see that:

*P*-value = 0.0262 < *α* = 0.05.

Thus, **we reject the null hypothesis**, *H*_{0}: *p* = 0.40. Our sample data provide significant evidence that the population proportion is not 0.40, and in fact, is likely much less. This means that significantly fewer people had “a great deal” of confidence in public schools in the year 2005 compared with the year 1995.