# Fast food Accuracy| Confidence Interval and Hypothesis Testing for Population Proportion| Statistics

### In a recent study of drive-through orders at Burger King, they found out that 365 were accurate and 71 were not accurate.a) Construct a 95% confidence interval for the population percentage of their drive-through order that were not accurate.b) If Burger King claims to have an accuracy of 85% on all drive-through orders, test the claim at the 0.05 levels of significance.

SOLUTION:

Part A

Compute for the value of the point estimate. There are $x=71$ successes (orders that were not accurate) out of $n=436$ drive-through orders in the study, so $\hat{p}=\frac{x}{n}=\frac{71}{436}$

Because we want a 95% confidence interval, we have $\alpha =1-0.95=0.05$, so $z_{\alpha /2}=z_{0.05/2}=z_{0.025}=1.96$

So, we have everything we need to solve for the lower and upper bounds of the confidence interval.

The lower bound is: $\hat{p}-z_{\alpha /2}\cdot \sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}}$ $=\frac{71}{436}-1.96\cdot \sqrt{\frac{\frac{71}{436}\left(1-\frac{71}{436}\right)}{436}}$ $=\frac{71}{436}-0.0347$ $=0.128\:or\:12.8\%$

The upper bound is: $\hat{p}+z_{\alpha /2}\cdot \sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}}$ $=\frac{71}{436}+1.96\cdot \sqrt{\frac{\frac{71}{436}\left(1-\frac{71}{436}\right)}{436}}$ $=\frac{71}{436}+0.0347$ $=\frac{71}{436}+0.0347$ $=0.198\:or\:19.8\%$

Therefore, we are 95% confident that the percentage of Burger King’s drive-through orders that were not accurate is between 12.8% and 19.8%.

Part B

The null and alternative hypothese are: $H_0:\:p=0.85$ $H_1:\:p\ne 0.85$

The sample proportion is $\:\hat{p}=\frac{365}{436}=0.8372$

The test statistic is $z_0=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}$ $z_0=\frac{0.8372-0.85}{\sqrt{\frac{0.85\left(1-0.85\right)}{436}}}$ $z_0=-0.75$

Because this is a two-tailed test, we determine the critical value at $\alpha /2=0.025$ level of significance to be $z_{0.025}=\pm 1.96$. The critical region is shown in the figure below. Since the test statistic $\left(z_0=-0.75\right)$ is not in the rejection region, we FAIL TO REJECT the null hypothesis.

There is sufficient evidence at the $\alpha =0.05$ level of significance to conclude that Burger King’s accuracy in their drive-through orders is 85%.