Answer to Question #180928 in Statistics and Probability for Richard Sasunathan

Question #180928

Question 1

An airline company wishes to know the proportion of business class travelers flying the Kuala Lumpur-to-Hong Kong route. In a random sample of 350 passengers, 190 are business class passengers.

 

a. Determine the point estimate of the true proportion of business class passengers.

b. Construct a 95% confidence interval estimate of the average waiting time for all customers.

c. Construct a 99% confidence interval estimate of the average waiting time for all customers.

d. Referring to b) and c), what will happen to the width of confidence interval?



1
Expert's answer
2021-04-15T06:30:38-0400

a. The point estimate of the true proportion of business class passengers

"\\hat{p}=\\dfrac{X}{N}=\\dfrac{190}{350}=\\dfrac{19}{35}\\approx0.542857"



b. The critical value for "\\alpha=0.05" is "z_c=z_{1-\\alpha\/2}=1.96."

The corresponding confidence interval is computed as shown below:


"CI(proportion)"

"=(\\hat{p}-z_c\\sqrt{\\dfrac{\\hat{p}(1-\\hat{p})}{n}},\\hat{p}+z_c\\sqrt{\\dfrac{\\hat{p}(1-\\hat{p})}{n}})"

"=(\\dfrac{19}{35}-1.96\\sqrt{\\dfrac{\\dfrac{19}{35}(1-\\dfrac{19}{35})}{350}},"

"\\dfrac{19}{35}+1.96\\sqrt{\\dfrac{\\dfrac{19}{35}(1+\\dfrac{19}{35})}{350}})"


"=(0.491,0.595)"

Therefore, based on the data provided, the 95% confidence interval for the population proportion is "0.491<x<0.595," which indicates that we are 95% confident that the true population proportion "p" is contained by the interval "(0.491,0.595)."


c. The critical value for "\\alpha=0.01" is "z_c=z_{1-\\alpha\/2}=2.576."

The corresponding confidence interval is computed as shown below:


"CI(proportion)"

"=(\\hat{p}-z_c\\sqrt{\\dfrac{\\hat{p}(1-\\hat{p})}{n}},\\hat{p}+z_c\\sqrt{\\dfrac{\\hat{p}(1-\\hat{p})}{n}})"

"=(\\dfrac{19}{35}-2.576\\sqrt{\\dfrac{\\dfrac{19}{35}(1-\\dfrac{19}{35})}{350}},"




"\\dfrac{19}{35}+2.576\\sqrt{\\dfrac{\\dfrac{19}{35}(1+\\dfrac{19}{35})}{350}})"


"=(0.474,0.611)"

Therefore, based on the data provided, the 99% confidence interval for the population proportion is "0.474<x<0.611," which indicates that we are 99% confident that the true population proportion "p" is contained by the interval "(0.474,0.611)."


d. If you want a higher level of confidence, that confidence interval will be wider.

Increasing the confidence will increase the margin of error resulting in a wider interval.



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