In a random sample of 1,000 exams, the average score was 500 points with a standard deviation of 80 points. Find the interval about the sample mean such that the probability is 0.90 that the mean number lies within the interval.
90% = mean ± 1.645 SEm
SEm = SD/√n
I'll let you do the calculations.
To find the interval about the sample mean such that the probability is 0.90, we can use the concept of confidence interval.
A confidence interval is a range of values that is likely to contain the true population parameter, in this case, the mean score of the exams.
Let's calculate the confidence interval using the given information:
1. Determine the level of confidence: In this case, the probability is given as 0.90, which corresponds to a 90% confidence level.
2. Find the critical value: The critical value is based on the confidence level and the distribution of the data. Since we have a large sample size (1,000 exams), we can assume a normal distribution. For a 90% confidence level, we need to find the z-score that corresponds to a cumulative probability of 0.95 (since we need to split the remaining 10% into the tails of the distribution). Using a standard normal distribution table or calculator, the z-score is approximately 1.645.
3. Calculate the standard error: The standard error represents the standard deviation of the sampling distribution of the mean and is calculated by dividing the standard deviation of the population (80 points) by the square root of the sample size (√1000).
Standard Error = Standard Deviation / √Sample Size
= 80 / √1000
= 2.527
4. Compute the margin of error: The margin of error represents the amount we need to add or subtract from the sample mean to obtain the interval estimate. It is calculated by multiplying the critical value by the standard error.
Margin of Error = Critical Value * Standard Error
= 1.645 * 2.527
≈ 4.155
5. Calculate the confidence interval: The confidence interval is computed by subtracting and adding the margin of error to the sample mean.
Confidence Interval = Sample Mean ± Margin of Error
= 500 ± 4.155
Therefore, the interval about the sample mean such that the probability is 0.90 (or 90% confidence interval) is approximately (495.845, 504.155).