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A random sample of the test scores of 100 applicants for college entry at a large junior college showed a mean score of 72.6. Determine a 99%, 95%, and 90% confidence interval for the m
mean score of all applicants at the
junior college. Assume the standard deviation of test scores is 10.5.
PTS Confidence interval
4 1. 99% confidence level x
4 2. 95% confidence level x
4 3. 90% confidence level x
To determine the confidence intervals for the mean score of all applicants at the junior college, we can use the formula:
Confidence interval = sample mean ± (critical value * standard deviation / square root of sample size)
Let's calculate the confidence intervals for the given confidence levels:
1. 99% confidence level:
To find the critical value for a 99% confidence level, we need to look up the value in a standard normal distribution table. For a 99% confidence level, the critical value is approximately 2.626.
Confidence interval = 72.6 ± (2.626 * 10.5 / √100)
Confidence interval = 72.6 ± (2.626 * 10.5 / 10)
Confidence interval = 72.6 ± (2.626 * 1.05)
Confidence interval = 72.6 ± 2.7543
Confidence interval = (69.8457, 75.3543)
So, the 99% confidence interval for the mean score of all applicants at the junior college is (69.8457, 75.3543).
2. 95% confidence level:
For a 95% confidence level, the critical value is approximately 1.96.
Confidence interval = 72.6 ± (1.96 * 10.5 / √100)
Confidence interval = 72.6 ± (1.96 * 10.5 / 10)
Confidence interval = 72.6 ± (1.96 * 1.05)
Confidence interval = 72.6 ± 2.058
Confidence interval = (70.542, 74.658)
The 95% confidence interval for the mean score of all applicants at the junior college is (70.542, 74.658).
3. 90% confidence level:
For a 90% confidence level, the critical value is approximately 1.645.
Confidence interval = 72.6 ± (1.645 * 10.5 / √100)
Confidence interval = 72.6 ± (1.645 * 10.5 / 10)
Confidence interval = 72.6 ± (1.645 * 1.05)
Confidence interval = 72.6 ± 1.727
Confidence interval = (70.873, 74.327)
The 90% confidence interval for the mean score of all applicants at the junior college is (70.873, 74.327).
Therefore, the confidence intervals for the mean score of all applicants at the junior college are as follows:
1. 99% confidence level: (69.8457, 75.3543)
2. 95% confidence level: (70.542, 74.658)
3. 90% confidence level: (70.873, 74.327)