Question

A new Mexican restaurant concept is being market tested. One hundred consumers who had visited the restaurant were surveyed and asked whether they would give the restaurant 5 stars. The responses were collected, as shown below:

	Response	Frequency
	5 stars	60
	Less than 5 stars	40
		100

PART 1: What would be the point estimate for the proportion of people who will give the restaurant 5 stars (to 2 decimal places)?

PART 2: What would be the margin of error for a 95% confidence interval estimate for the proportion of people who will give the restaurant 5 stars (to 3 decimal places)?

PART 3: What would be the LOW END for the 95% confidence interval estimate for the proportion of people who will give the restaurant 5 stars (to 3 decimal places)? (In other words, if the interval estimate is between a and b, what is the value of a?)

PART 4: What would be the HIGH END for the 95% confidence interval estimate for the proportion of people who will give the restaurant 5 stars (to 3 decimal places)? (In other words, if the interval estimate is between a and b, what is the value of b?)

PART 5: With a 0.95 probability, what sample size needs to be collected, to provide a margin of error of 0.08 or less?

Answer 1

1)

Solution
sample proportion, = 0.6

2)

sample size, n = 100
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.6 * (1 - 0.6)/100) = 0.049

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.049
ME = 0.096

3)

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.6 - 1.96 * 0.049 , 0.6 + 1.96 * 0.049)
CI = (0.504 , 0.696)

low end = 0.504

4)

High end = 0.696

5)

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.08

The provided estimate of proportion p is, p = 0.6
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.6*(1 - 0.6)*(1.96/0.08)^2
n = 144.06

Therefore, the sample size needed to satisfy the condition n >= 144.06 and it must be an integer number, we conclude that the minimum required sample size is n = 145
Ans : Sample size, n = 145 or 144