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.
4.90 (round to two decimal places as needed.)
what is the effect of the sample size on the margin of error? choose the correct answer below
○ a. the larger sample size increases the margin of error
○ b. the margin of error is the same for all sample sizes
○ c. the larger sample size decreases the margin of error
○ d. there is no correlation between the sample size and the margin of error

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), \(s = 100\) (sample standard deviation).

Step2: Analyze the relationship between \(n\) and \(E\)

We can rewrite the formula as \(E\propto\frac{1}{\sqrt{n}}\) (since \(z\) and \(s\) are constant in this problem). When \(n\) increases, \(\sqrt{n}\) increases.
For example, when \(n_1 = 400\), \(\sqrt{n_1}=20\); when \(n_2 = 1600\), \(\sqrt{n_2} = 40\). As \(n\) goes from \(400\) to \(1600\) (increases), \(E\) goes from \(9.80\) to \(4.90\) (decreases)

Answer:

C. The larger sample size decreases the margin of error.