a)A university planner wants to determine the proportion of spring semester students who will attend summer school. Suppose the university would like a 0.90 probability that the sample proportion is within 0.195 or less of the population proportion.What is the smallest sample size to meet the required precision? (There is no estimation for the sample proportion.) (Enter an integer number.)
b)A university planner wants to determine the proportion of spring semester students who will attend summer school. She surveys 31 current students discovering that 19 will return for summer school.At 90% confidence, compute the margin of error for the estimation of this proportion.
c)A university planner wants to determine the proportion of spring semester students who will attend summer school. She surveys 36 current students discovering that 16 will return for summer school.At 90% confidence, compute the lower bound of the interval estimate for this proportion.
a)
The following information is provided,
Significance Level, α = 0.1, Margin of Error, E = 0.195
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)*(1.64/0.195)^2
n = 17.68
Therefore, the sample size needed to satisfy the condition n
>= 17.68 and it must be an integer number, we conclude that the
minimum required sample size is n = 18
Ans : Sample size, n = 18
b)
sample proportion, = 0.6129
sample size, n = 31
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.6129 * (1 - 0.6129)/31) = 0.0875
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 * 0.0875
ME = 0.1435
v)
sample proportion, = 0.4444
sample size, n = 36
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.4444 * (1 - 0.4444)/36) = 0.0828
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 * 0.0828
ME = 0.1358
CI = (pcap - z*SE, pcap + z*SE)
CI = (0.4444 - 1.64 * 0.0828 , 0.4444 + 1.64 * 0.0828)
CI = (0.309 , 0.58)
Lower bound = 0.309
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