Determine the 90% confidence interval for a sample, taken from a normally distributed population, that has a mean of 685, a sample standard deviation of 74, and a sample size of 29
90% interval = mean ± 1.645 SEm
SEm = SD/√n
To determine the 90% confidence interval for a sample with a known mean, population standard deviation, and sample size, you can use the formula:
Confidence Interval = Sample Mean ± Margin of Error
Step 1: Calculate the Margin of Error
The margin of error (E) is calculated using the formula:
Margin of Error = Z * (Sample Standard Deviation / √(Sample Size))
Where:
Z is the Z-score corresponding to the desired confidence level. For a 90% confidence level, the Z-score is 1.645.
Sample Standard Deviation is given as 74.
Sample Size is given as 29.
Margin of Error = 1.645 * (74 / √(29))
Step 2: Calculate the Confidence Interval
Now that we have the margin of error, we can calculate the confidence interval by adding and subtracting the margin of error from the sample mean.
Confidence Interval = Sample Mean ± Margin of Error
Confidence Interval = 685 ± (1.645 * (74 / √(29)))
Step 3: Simplify the Confidence Interval
Calculating the confidence interval:
Confidence Interval = 685 ± 1.645 * (74 / √(29))
Using a calculator or computer software, we can calculate the margin of error and simplify the confidence interval:
Confidence Interval = 685 ± 16.360
Therefore, the 90% confidence interval for the given sample is (668.640, 701.360).