The population standard deviation (σ) for a standardized achievement test that is normally distributed is 8.
a.) Calculate the standard error of the mean if you draw a random sample of 50 test scores. Show your work for full credit . . .
b.) Your sample mean is calculated to be 77. Compute the 95% confidence interval for the mean. Show your work!
c.) Interpret the meaning of the 95% confidence interval in one sentence.
Here are a few hints to get you started:
a). Standard error of the mean = standard deviation divided by the square root of the sample size
b). CI95 = mean ± 1.96(sd/√n)
...where ± 1.96 represents the 95% interval using a z-table, sd = standard deviation, and n = sample size
c). You are 95% confident that the population mean is within the interval.
a.) To calculate the standard error of the mean (SE), you need to divide the population standard deviation (σ) by the square root of the sample size (n). In this case, the population standard deviation is given as 8, and the sample size is 50, so the formula for calculating the standard error of the mean is:
SE = σ / √n
Substituting the values, we have:
SE = 8 / √50
To simplify, we can approximate the value of the square root of 50 as 7.07:
SE ≈ 8 / 7.07 ≈ 1.13
Therefore, the standard error of the mean for a random sample of 50 test scores is approximately 1.13.
b.) To compute the 95% confidence interval for the mean, you need to use the formula:
Confidence Interval = Sample mean ± Margin of Error
The margin of error is calculated by multiplying the standard error (SE) by the critical value corresponding to the desired level of confidence. Since we want a 95% confidence interval, the critical value is obtained from the Z-table or a statistical software, which is approximately 1.96.
Margin of Error = Z * SE
Substituting the values, we have:
Margin of Error = 1.96 * 1.13 ≈ 2.22
The sample mean is given as 77, so we can calculate the confidence interval as:
Confidence Interval = 77 ± 2.22
This gives a lower bound of 74.78 and an upper bound of 79.22.
Therefore, the 95% confidence interval for the mean is approximately 74.78 to 79.22.
c.) The 95% confidence interval means that if this experiment were repeated many times, we would expect that 95% of the calculated confidence intervals would contain the true population mean. In other words, we are 95% confident that the true population mean lies within the computed interval of 74.78 to 79.22.