A simple random sample of 60 items resulted in a sample mean of 76. The population standard deviation is 12.
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).
To compute the confidence intervals for the population mean, we'll be using the formula:
Confidence Interval = sample mean ± margin of error
The margin of error is determined by the confidence level, the sample standard deviation, and the sample size. For both questions, we'll be using a 95% confidence level.
a. For a sample size of 60 items, with a sample mean of 76 and a population standard deviation of 12, we can find the margin of error as follows:
Standard Error = population standard deviation / square root of sample size
= 12 / sqrt(60)
≈ 1.55
To calculate the margin of error, we multiply the standard error by the critical value associated with a 95% confidence level. Since the sample size is relatively large (n > 30), we can use the Z-distribution and find the critical value using a Z-table or calculator. The critical value for a 95% confidence level is approximately 1.96.
Margin of Error = standard error * critical value
= 1.55 * 1.96
≈ 3.03
Now we can compute the confidence interval:
Confidence Interval = sample mean ± margin of error
= 76 ± 3.03
≈ 72.97 to 79.03 (rounded to one decimal place)
Therefore, the 95% confidence interval for the population mean is approximately 72.97 to 79.03.
b. For a sample size of 120 items with the same sample mean of 76 and a population standard deviation of 12, we'll follow the same steps as before. However, since the sample size has now doubled, the standard error and margin of error will change.
Standard Error = population standard deviation / square root of sample size
= 12 / sqrt(120)
≈ 1.10
Critical Value for a 95% confidence level = 1.96, as before.
Margin of Error = standard error * critical value
= 1.10 * 1.96
≈ 2.16
Confidence Interval = sample mean ± margin of error
= 76 ± 2.16
≈ 73.84 to 78.16 (rounded to two decimal places)
Therefore, the 95% confidence interval for the population mean, when the sample size is 120, is approximately 73.84 to 78.16.
To compute the confidence intervals, we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / √sample size)
where the critical value depends on the level of confidence.
a. Compute the 95% confidence interval for the population mean:
Step 1: Find the critical value for a 95% confidence level. We can use a Z-table or a calculator. Since the sample size is large (n > 30), we can use the Z distribution. For a 95% confidence level, the critical value is approximately 1.96.
Step 2: Plug in the values into the formula.
Confidence Interval = 76 ± (1.96) * (12 / √60)
Step 3: Calculate the confidence interval.
Confidence Interval = 76 ± (1.96) * (12 / 7.75) ≈ 76 ± 3.01
The 95% confidence interval for the population mean is approximately (72.99, 79.01).
b. Assume that the same sample mean was obtained from a sample of 120 items.
Step 1: The critical value remains the same since it is based on the confidence level which is still 95%. The critical value is approximately 1.96.
Step 2: Plug in the values into the formula.
Confidence Interval = 76 ± (1.96) * (12 / √120)
Step 3: Calculate the confidence interval.
Confidence Interval = 76 ± (1.96) * (12 / 10.95) ≈ 76 ± 2.15
The 95% confidence interval for the population mean is approximately (73.85, 78.15).