3.3 Single population proportion

• The population parameter $p$ of the Binomial distribution can also be tested using similar procedure

$$\begin{equation} H_0:~~p=p_0 \tag{3.9} \end{equation}$$

• Assumed population proportion $p_0$ is the value that we think is true. If sample data shows that this is false, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.

• Three alternative hypothesis are possible:

$$\begin{align} H_1:&~~p \ne p_0 &\text{two-tailed test} \\ \\ H_1:&~~p < p_0 &\text{left-tailed test} \\ \\ H_1:&~~p > p_0 &\text{right-tailed test} \\ \tag{3.10} \end{align}$$

• As the binomial distribution of a sample proportion $\hat{p}$ can be approximated by the normal distribution, the test statistic follows a standard normal distribution:

$$\begin{align} Z&=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \\ \\ \hat{p}&=\frac{x}{n} \\ \tag{3.11} \end{align}$$

Example 3.5 Suppose a consumer group suspects that the proportion of households that have three or more cell phones is $30\%$. A cell phone company has reason to believe that the proportion is not $30\%$. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey $150$ households with the result that $43$ of them have three or more cell phones. Level of significance is $5%$.

Example 3.6 Test the hypothesis that more than $50\%$ of the companies are listed on the stock exchange. Level of significance is $5\%$.