A simple random sample of 60 items resulted in a sample mean of 96. The population standard deviation is 16.

a. Compute the 95% confidence interval for the population mean (to 1 decimal).
( , )

b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean (to 2 decimals).
( , )
c. What is the effect of a larger sample size on the margin of error, does it increase, decrease,stays the same, or it cannot be determined?

NOTE: I don't know if it is right.
a. I got (194.2,202.3)
b. I got (137.35, 143.08)
c. It decrease

5 answers

A)

z = 1.96
n = 60
x-bar = 96
S = 16
SE = σ/√n = 16/√60
xbar ± z*s/√n
96 ± 4.048
( 92.0, 100.0)

B)
z = 1.96
SE = σ/√n = 16/√120
96 - 2.867, 96 + 2.867
(93.1, 98.9)

Only c is correct

Can you help me please?

thanks

I did not know how to do it.

To calculate the confidence intervals, we can use the formula:

Confidence Interval = sample mean ± (critical value * standard deviation / square root of sample size)

a. For a 95% confidence interval, the critical value is found using the z-table. Since it is not provided, we will use a standard normal distribution which corresponds to a 95% confidence level. The critical value for a 95% confidence level is approximately 1.96.

Plugging in the values:
Sample mean = 96
Standard deviation = 16
Sample size = 60

Confidence Interval = 96 ± (1.96 * 16 / sqrt(60))

Solving this formula:
Confidence Interval = (96 - 3.1, 96 + 3.1)
Confidence Interval ≈ (92.9, 99.1)

So, the 95% confidence interval for the population mean is (92.9, 99.1) (to 1 decimal place).

b. When the sample size increases to 120, the margin of error decreases, leading to a narrower confidence interval. Using the same approach as in part a, we obtain:

Sample mean = 96
Standard deviation = 16
Sample size = 120

Confidence Interval = 96 ± (1.96 * 16 / sqrt(120))

Solving this formula:
Confidence Interval = (96 - 4.6, 96 + 4.6)
Confidence Interval ≈ (91.4, 100.6)

So, the 95% confidence interval for the population mean with a sample size of 120 is (91.4, 100.6) (to 2 decimal places).

c. When the sample size increases, the margin of error decreases. This means that with a larger sample size, the confidence interval becomes narrower, providing a more precise estimate of the population mean. The effect of a larger sample size is to decrease the margin of error, resulting in a more precise estimate.