Question

please do these 2 problem step by step and how how you get z core

SAT scores are normally distributed with a mean of 1,500 and a standard deviation of 300. An administrator at a college is interested in estimating the average SAT score of first-year students. If the administrator would like to limit the margin of error of the 86% confidence interval to 15 points, how many students should the administrator sample? Make sure to give a whole number answer.

You are interested in estimating the the mean age of the citizens living in your community. In order to do this, you plan on constructing a confidence interval; however, you are not sure how many citizens should be included in the sample. If you want your sample estimate to be within 5 years of the actual mean with a confidence level of 96%, how many citizens should be included in your sample? Assume that the standard deviation of the ages of all the citizens in this community is 25 years.

Answer 1

The following information is provided,
Significance Level, α = 0.14, Margin or Error, E = 15, σ = 300

The critical value for significance level, α = 0.14 is 1.48.

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 = (1.48 * 300/15)^2
n = 876.16

Therefore, the sample size needed to satisfy the condition n >= 876.16 and it must be an integer number, we conclude that the minimum required sample size is n = 877
Ans : Sample size, n = 877 or 876

2)

The following information is provided,
Significance Level, α = 0.04, Margin or Error, E = 5, σ = 25

The critical value for significance level, α = 0.04 is 2.05.

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 = (2.05 * 25/5)^2
n = 105.06

Therefore, the sample size needed to satisfy the condition n >= 105.06 and it must be an integer number, we conclude that the minimum required sample size is n = 106
Ans : Sample size, n = 106 or 105

### if you take z value upto 3 decimal answer would be change