Answer
The range of population proportion increases as null hypothesis is not getting rejected for any value of proportion.
Work Step by Step
There are 53 out of 100 individuals who prefer Pepsi. The sample proportion is calculated as
\[\begin{align}
& \hat{p}=\frac{x}{n} \\
& =\frac{53}{100} \\
& =0.53
\end{align}\]
The confidence interval of the one sample proportion test is provided as
\[\hat{p}\pm {{z}_{{0.01}/{2}\;}}\times \sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}}\]
The critical value of z at 0.01 significance level for two-tailed test is calculated in Excel.
-2.5758
According to the data, the confidence interval at 95% level of significance is as follows:
\[\begin{align}
& \hat{p}\pm {{z}_{{0.01}/{2}\;}}\times \sqrt{\frac{\hat{p}\left( 1-\hat{p} \right)}{n}} \\
& 0.53\pm 2.56\times \sqrt{\frac{0.53\left( 1-0.53 \right)}{100}} \\
& 0.53\pm 0.1277 \\
& 0.4023,0.6577
\end{align}\]
The 99% confidence interval is \[\left[ 0.4023,0.6577 \right]\]. So, all the proportions that are 0.42 to 0.64 are lying in the interval. So, the null hypothesis does not get rejected for any value. The range of the population proportion increases as all lie in an interval.
