A simple random sample of 60 items resulted in a sample mean of 80. The population standard deviation is 15.

a) Compute the 95 % confidence interval for the population mean.
b) Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean.
c)What is the effect of a larger sample size on the margin of error?

I would really like to know this answer, i am not good at doing the calculations....can someone help please?

How do you get 1.96

To compute the confidence intervals, we'll use the formula:

Confidence Interval = sample mean ± (critical value) * (standard deviation / √n)

Where:
- Sample mean is the mean of the sample (80 in this case)
- Critical value is obtained from the Z-table for the chosen confidence level (95% confidence level in this case)
- Standard deviation is the population standard deviation (15 in this case)
- n is the sample size (60 for part a, and 120 for part b)

Let's solve each part step by step:

a) Compute the 95% confidence interval for the population mean (sample size = 60):
First, we need to find the critical value corresponding to a 95% confidence level. For a two-tailed test, the critical value is 1.96.

Confidence Interval = 80 ± (1.96) * (15 / √60)

Using a calculator or software, calculate the value inside the parentheses:
(1.96) * (15 / √60) ≈ 4.86

Therefore, the confidence interval is:
Confidence Interval = 80 ± 4.86
Confidence Interval = (75.14, 84.86)

b) Compute the 95% confidence interval for the population mean (sample size = 120):
The formula remains the same, but with a different sample size.

Confidence Interval = 80 ± (1.96) * (15 / √120)

Calculate the value inside the parentheses:
(1.96) * (15 / √120) ≈ 2.71

Therefore, the confidence interval is:
Confidence Interval = 80 ± 2.71
Confidence Interval = (77.29, 82.71)

c) The effect of a larger sample size on the margin of error:
As we can see from the calculations, as the sample size increases (from 60 to 120 in this case), the margin of error decreases. In other words, with a larger sample size, the confidence interval becomes narrower. This happens because a larger sample size provides more information about the population, resulting in a more precise estimate of the population mean.

a) 95% = mean ± 1.96 SD

b) 95% = mean ± 1.96 SEm

SEm = SD/√(n-1)

c) margin of error decreases.