Question

Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95​% confidence that the sample mean is plus or minus3 IQ points of the true mean. Assume that the standard deviation is 15 and determine the required sample size.

2)

In a survey of

2,416 adults, 1,876 reported that​ e-mails are easy to​ misinterpret, but only 1,231 reported that telephone conversations are easy to misinterpret. Complete parts​ (a) through​ (c) below.

a. Construct a​ 95% confidence interval estimate for the population proportion of adults who report that​ e-mails are easy to misinterpret.

Answer 1

#1.
The following information is provided,
Significance Level, α = 0.05, Margin or Error, E = 3, σ = 15

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 mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (1.96 * 15/3)^2
n = 96.04

Therefore, the sample size needed to satisfy the condition n >= 96.04 and it must be an integer number, we conclude that the minimum required sample size is n = 97
Ans : Sample size, n = 97

#2.
sample proportion, = 0.5095
sample size, n = 2416
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.5095 * (1 - 0.5095)/2416) = 0.0102

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

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.5095 - 1.96 * 0.0102 , 0.5095 + 1.96 * 0.0102)
CI = (0.4895 , 0.5295)