Question

The manager of the dairy section of a large supermarket took a random sample of 250 egg cartons and found that 40 cartons had at least one broken egg. (a) Let p be the proportion of egg cartons with at least one broken egg out of the population of all egg cartons stocked by this store. Find a point estimate for p. (b) Find a 90% confidence interval for p. How many egg cartons would the manager need to examine to be 90% sure that the point estimate of the proportion of cartons with at least one broken egg is within 3% of the population proportion p? Answer this question: (c) assuming a prior estimate of p equal to 0.16, (d) assuming no prior point estimate.

Answer 1

a)

point estimate = sample proportion, = 0.16

b)

sample size, n = 250
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.16 * (1 - 0.16)/250) = 0.0232

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, Zc = Z(α/2) = 1.64

Margin of Error, ME = zc * SE
ME = 1.64 * 0.0232
ME = 0.038

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.16 - 1.64 * 0.0232 , 0.16 + 1.64 * 0.0232)
CI = (0.122 , 0.198)

c)

The following information is provided,
Significance Level, α = 0.1, Margin of Error, E = 0.03

The provided estimate of proportion p is, p = 0.16
The critical value for significance level, α = 0.1 is 1.64.

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 = 0.16*(1 - 0.16)*(1.64/0.03)^2
n = 401.65

Therefore, the sample size needed to satisfy the condition n >= 401.65 and it must be an integer number, we conclude that the minimum required sample size is n = 402
Ans : Sample size, n = 402

d)

The following information is provided,
Significance Level, α = 0.1, Margin of Error, E = 0.03

The provided estimate of proportion p is, p = 0.5
The critical value for significance level, α = 0.1 is 1.64.

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 = 0.5*(1 - 0.5)*(1.64/0.03)^2
n = 747.11

Therefore, the sample size needed to satisfy the condition n >= 747.11 and it must be an integer number, we conclude that the minimum required sample size is n = 748
Ans : Sample size, n = 748 or 747