Question

The owner of a restaurant is reviewing customer complaints. In a random sample of 227 complaints, 57 complaints were about the slow speed of the service. Create a 95% confidence interval for the proportion of complaints that were about the slow speed of the service. Use Excel to create the confidence interval, rounding to four decimal places.

Answer 1

Answer:

$0.1375$ to $0.3624$

Step-by-step explanation:

Given :The owner of a restaurant is reviewing customer complaints. In a random sample of 227 complaints, 57 complaints were about the slow speed of the service.

To Find :Create a 95% confidence interval for the proportion of complaints that were about the slow speed of the service.

Solution:

n = 227

x = 57

Formula of confidence for proportion: $\widecap{p}-z_{\frac{\alpha}{2}}\sqrt{\frac{\widecap{p}\widecap{q}}{n}}$ to  $\widecap{p}+z_{\frac{\alpha}{2}}\sqrt{\frac{\widecap{p}\widecap{q}}{n}}$

$\widecap{p}=\frac{x}{n}$

$\widecap{p}=\frac{57}{227}$

$\widecap{p}=0.25$

$\widecap{q}=1-\widecap{p}$

$\widecap{q}=1-0.25$

$\widecap{q}=0.75$

z at 95% is 1.96

Substitute the values in the formula :

Confidence for proportion: $0.25-1.96\sqrt{\frac{0.25 \times 0.75}{57}}$ to $0.25+1.96\sqrt{\frac{0.25 \times 0.75}{57}}$

Confidence for proportion: $0.1375$ to $0.3624$

Hence 95% confidence interval for the proportion of complaints that were about the slow speed of the service is  $0.1375$ to $0.3624$

Answer 2

Answer:

0.1947-0.3075

The answer for the margin of error, rounded to four decimal places, is z⋆p^(1−p^)n−−−−−−−−√≈0.0564. The confidence interval for the population proportion has a lower limit of A4−A7=0.1947 and an upper limit of A4+A7=0.3075. Thus, the 95% confidence interval for the population proportion of complaints that were about the slow speed of the service, based on this sample, is approximately (0.1947, 0.3075).