Salaries of 50 college graduates who took a statistics course in college have a mean, x of $68,200. Assuming a standard deviation, σ, of $12,279, construct a 95% confidence interval for estimating the population mean μ.
95% = mean ± Z(SD)
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.025) and its Z score. Insert data into above equation and solve.
To construct a 95% confidence interval for estimating the population mean μ, we can use the formula:
Confidence Interval = x ± (Z * (σ / √n))
Where:
- x represents the sample mean
- Z represents the z-score corresponding to the desired level of confidence (in this case, a 95% confidence level)
- σ represents the population standard deviation
- n represents the sample size
In this case, the sample mean (x) is $68,200, the standard deviation (σ) is $12,279, and the sample size (n) is 50.
Step 1: Find the z-score corresponding to a 95% confidence level.
Since we want a 95% confidence level, we divide the desired confidence level by 2 to find the tail probability on each side. Our desired tail probability is (1 - (0.95))/2 = 0.025. We can then look up the z-score that corresponds to this tail probability in a standard normal distribution table or use a calculator. In this case, the z-score for a 95% confidence level is approximately 1.96.
Step 2: Calculate the margin of error.
The margin of error is calculated by multiplying the z-score by the standard deviation divided by the square root of the sample size. In this case, the margin of error is 1.96 * ($12,279 / √50) = $3429.71.
Step 3: Construct the confidence interval.
The confidence interval is calculated by adding and subtracting the margin of error from the sample mean. In this case, the confidence interval is $68,200 ± $3429.71.
Therefore, the 95% confidence interval for estimating the population mean μ is ($64,770.29 - $71,629.71).