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Step-By-Step Example of a Confidence Interval for a Mean, sigma known (used Ex 8.2)
Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of 3 points. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68.
Find a 90% confidence interval for the true (population) mean of statistics exam scores.
| Guidelines | Example |
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| Plan: State what we need to know. | We are asked to find a 90% confidence interval for the mean exam score, μ, of statistics students. We have a sample of 68 students. We know the population standard deviations is 3. |
| Model: Think about the assumptions and check the conditions. |
Randomization Condition: The sample is a random sample. Independence Assumption: It is reasonable to think that the exam scores of 36 randomly selected students are independent. 10% Condition: I assume the statistic student population is over 360 students, so 36 students is less than 10% of the population. Sample Size Condition: Since the distribution of the stress levels is normal, my sample of 36 students is large enough. |
| State the parameters and the sampling model | The conditions are satisfied and σ is known, so we will use a confidence interval for a mean with known standard deviation.We need the sample mean and Margin of Error (ME).
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Mechanics: CL = 0.90, so α = 1-CL = 1-0.90 = 0.10. ;
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| Conclusion: Interpret your result in the proper context, and relate it to the original question. | I am 90% confident that the interval from 67.1775% to 68.8225% contains the true mean score of all the statistics exams. 90% of all confidence intervals constructed in this way contain the true mean statistics exam score. |
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