A random sample of 100 GMAT scores has a mean of 500. Assume that GMAT scores have a population standard deviation of 120. Construct a 95% confidence interval estimate of the mean GMAT scores. (15 pts)
What is the sample size for the conference interval?
What is the point estimate for the conference interval?
What is the numerical value of the variable from the above distribution?
To construct a confidence interval estimate of the mean GMAT scores, we'll use the formula:
Confidence interval = sample mean ± (critical value * standard error)
Step 1: Determine the sample size for the confidence interval.
The sample size for the confidence interval is given in the problem as 100.
Step 2: Calculate the point estimate for the confidence interval.
The point estimate is the sample mean, which is also given in the problem as 500.
Step 3: Determine the critical value.
We need to determine the critical value using the confidence level of 95%. Since the sample size is large (n > 30), we can use a normal distribution and look up the z-value associated with a 95% confidence level. The critical value for a 95% confidence level is approximately 1.96.
Step 4: Calculate the standard error.
The standard error is calculated by dividing the population standard deviation by the square root of the sample size.
Standard error = population standard deviation / √sample size
= 120 / √100
= 12.
Step 5: Calculate the confidence interval.
Confidence interval = sample mean ± (critical value * standard error)
= 500 ± (1.96 * 12)
= 500 ± 23.52.
Therefore, the confidence interval estimate of the mean GMAT scores is 476.48 - 523.52.
To answer your specific questions:
1. The sample size for the confidence interval is 100.
2. The point estimate for the confidence interval is 500.
3. The numerical value of the variable from the above distribution is 23.52.
To construct a confidence interval estimate of the mean GMAT scores, we use the formula:
Confidence Interval = Point Estimate ± Margin of Error
To calculate the confidence interval, we need to find the sample size, the point estimate, and the margin of error.
1. Sample Size:
The sample size for the confidence interval is given as 100. This means that the sample contains 100 GMAT scores.
2. Point Estimate:
The point estimate for the confidence interval is the sample mean, which is given as 500. This means that the average GMAT score in the sample is 500.
3. Margin of Error:
To calculate the margin of error, we need to consider the standard deviation of the population (120), the sample size (100), and the desired confidence level (95%).
The formula to calculate the margin of error is given by:
Margin of Error = Critical Value * (Standard Deviation / Square Root of Sample Size)
To find the critical value, we refer to the z-table for a 95% confidence level, which corresponds to a z-score of approximately 1.96.
Margin of Error = 1.96 * (120 / √100)
Margin of Error = 1.96 * (120 / 10)
Margin of Error = 1.96 * 12
Margin of Error ≈ 23.52
4. Confidence Interval:
Finally, we can construct the confidence interval using the point estimate and the margin of error:
Confidence Interval = 500 ± 23.52
Confidence Interval = (476.48, 523.52)
Therefore, the answers to the specific questions are:
1. Sample size for the confidence interval: 100
2. Point estimate for the confidence interval: 500
3. Numerical value of the variable from the above distribution: The variable in this case would be the GMAT scores, and it ranges from 476.48 to 523.52 in the given sample.