Question

find the margin of error for a 95% confidence interval for estimating the population mean when the sample standard deviation equals 100, with a sample size of (i) 400 and (ii) 1600. what is the effect of the sample size? (i) find the margin of error for a 95% confidence interval for estimating the population mean when the sample standard deviation equals 100 with a sample size of 400. 9.80 (round to two decimal places as needed.) (ii) find the margin of error for a 95% confidence interval for estimating the population mean when the sample standard deviation equals 100 with a sample size of 1600. (round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for margin of error

The formula for margin of error \(E = z\times\frac{s}{\sqrt{n}}\). For a 95% confidence interval, \(z = 1.96\) (approximate value from the standard normal distribution), \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step2: Calculate for \(n = 1600\) and \(s=100\)

Substitute \(z = 1.96\), \(s = 100\), and \(n = 1600\) into the formula.
First, calculate \(\sqrt{n}=\sqrt{1600}=40\).
Then, \(\frac{s}{\sqrt{n}}=\frac{100}{40} = 2.5\).
Finally, \(E=1.96\times2.5\).
\(E = 4.90\)

Answer:

(i) \(9.80\)
(ii) \(4.90\)
The effect of the sample size: As the sample size increases, the margin of error decreases.