Question

Suppose we are interested in finding a 95% confidence interval for the mean SAT Verbal score of students at a certain high school. Five students are sampled, and their SAT Verbal scores are 560, 500, 470, 660, and 640.

A. What is the standard error of the sample mean?
B. What is the 95% confidence interval for the population mean?

Answer 1

The standard error of the sample mean is 68.01. The 95% confidence interval for the population mean is (377.18, 754.82).

Step-by-step explanation:

A. To calculate the standard error of the sample mean, we need to calculate the standard deviation of the sample data first. The sample data consists of 5 students' SAT Verbal scores: 560, 500, 470, 660, and 640. To calculate the standard deviation, we follow these steps:

1. Calculate the mean of the sample scores: (560 + 500 + 470 + 660 + 640) / 5 = 566.
2. Subtract the mean from each individual score and square the result: (560 - 566)^2, (500 - 566)^2, (470 - 566)^2, (660 - 566)^2, (640 - 566)^2.
3. Calculate the sum of the squared differences: (560 - 566)^2 + (500 - 566)^2 + (470 - 566)^2 + (660 - 566)^2 + (640 - 566)^2 = 92780.
4. Divide the sum by the number of scores minus 1: 92780 / (5 - 1) = 23195.
5. Finally, take the square root of the result to get the standard deviation: sqrt(23195) ≈ 152.19.

Now, to calculate the standard error of the sample mean, we divide the standard deviation by the square root of the sample size: 152.19 / sqrt(5) ≈ 68.01.

B. The 95% confidence interval for the population mean can be calculated using the formula: (sample mean - margin of error, sample mean + margin of error). The margin of error is obtained by multiplying the standard error by the critical value corresponding to a 95% confidence level. For a sample size of 5, the critical value is 2.776.

So, the margin of error is 2.776 * 68.01 = 188.82. The sample mean is 566. Therefore, the 95% confidence interval for the population mean is (566 - 188.82, 566 + 188.82) ≈ (377.18, 754.82).