Twenty randomly selected students took the statistics final. If the sample mean was 88 and the ó =5.2, construct a 95% confidence interval for the mean score of the students.
Formula:
CI95 = mean ± 1.96(sd/√n)
With your data:
CI95 = 88 ± 1.96(5.2/√20)
Calculate for the interval.
To construct a confidence interval for the mean score of the students, we'll use the formula:
Confidence interval = sample mean ± (critical value) × (standard deviation / square root of sample size)
Step 1: Determine the critical value
Since we want a 95% confidence interval, we need to find the critical value that corresponds to a 95% confidence level. The critical value can be found using a standard normal distribution table or a calculator. For a 95% confidence level, the critical value is approximately 1.96.
Step 2: Calculate the standard error
The standard error is the standard deviation of the sample mean and can be calculated using the formula:
Standard error = standard deviation / square root of sample size
In this case, the standard deviation (σ) is given as 5.2 and the sample size (n) is 20. Therefore:
Standard error = 5.2 / √20 ≈ 1.16
Step 3: Calculate the confidence interval
Now we can plug the values into the confidence interval formula:
Confidence interval = 88 ± (1.96) × (1.16) ≈ 88 ± 2.27
Therefore, the 95% confidence interval for the mean score of the students is approximately 85.73 to 90.27.
Note: This means that we are 95% confident that the true mean score lies within this interval based on the given sample.