Sample with size n = 100 has mean = 30. Assuming the population standard deviation is 8, construct 95% confidence interval for population mean.
Formula:
CI95 = mean ± 1.96(sd/√n)
mean = 30
sd = 8
n = 100
Plug the values into the formula, then calculate to determine your confidence interval.
To construct a 95% confidence interval for the population mean, you can use the formula:
Confidence interval = sample mean ± (critical value) * (standard deviation / √n)
Step 1: Identify the critical value
Since we are constructing a 95% confidence interval, we need to find the critical value associated with a 95% confidence level. This critical value depends on the sample size and the desired confidence level.
For large sample sizes (n > 30), we can use the Z-table to find the critical value. For a 95% confidence level, the critical value is approximately 1.96.
Step 2: Calculate the standard error
The standard error measures how much the sample mean varies from the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size.
Standard Error = Population Standard Deviation / √n
In this case, the population standard deviation is given as 8, and the sample size is 100. So, the standard error is:
Standard Error = 8 / √100 = 8 / 10 = 0.8
Step 3: Calculate the confidence interval
Now we are ready to calculate the confidence interval using the formula:
Confidence interval = sample mean ± (critical value) * (standard error)
The sample mean is given as 30, the critical value is 1.96 (for a 95% confidence level), and the standard error is 0.8. Plugging in these values, we get:
Confidence interval = 30 ± (1.96) * (0.8)
Calculating the values, we get:
Confidence interval = 30 ± 1.568
So the 95% confidence interval for the population mean is (28.432, 31.568).