the average placement test score of a sample of 64 students out of 1000 registered at a community college is 70 with a standard deviation of 15. Construct a confidence interval with a 5% chance of error.
95% = mean ± 1.96 SEm
SEm = SD/√n
To construct a confidence interval, we'll use the formula:
Confidence interval = sample mean ± Margin of error
Where:
- Sample mean is the average placement test score (given as 70).
- Margin of error is calculated by multiplying the critical value by the standard error.
- Standard error is the standard deviation of the sample divided by the square root of the sample size.
- Critical value is found using the z-table or z-score calculator, given the desired confidence level.
Step 1: Find the critical value
Since we want to construct a confidence interval with a 5% chance of error, we need to find the critical value associated with a 95% confidence level. This is equivalent to a significance level (alpha) of 0.05. Looking up the critical value in the z-table, for a two-tailed test, we find it to be approximately 1.96.
Step 2: Calculate the standard error
The standard error is the standard deviation divided by the square root of the sample size. Given that the standard deviation is 15 and the sample size is 64, we can calculate the standard error:
Standard error = 15 / √64 ≈ 1.875
Step 3: Calculate the margin of error
The margin of error is the product of the critical value and the standard error:
Margin of error = 1.96 * 1.875 ≈ 3.675
Step 4: Construct the confidence interval
Finally, we can construct the confidence interval by adding and subtracting the margin of error to the sample mean:
Confidence interval = 70 ± 3.675
Therefore, the confidence interval for the average placement test score is approximately (66.325, 73.675) or from 66.325 to 73.675.