# Confidence intervals and variance analysis

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Let (X1, X2,..., Xn) be a sample, where each Xi is a random variable of normal distribution with mean mu and variance sigma². Let us suppose that n = 20, and sigma² = 9. An experiment has yielded the results (X1, X2,..., X20), and we have calculated that the empirical mean x20 = 2.09.

1) Give a confidence interval with level of confidence 90% for mu.

2) How big would the sample have to be for the interval to be half as long?

3) Let us now suppose the variance is not known. Knowing that counting from i=1 to 20 (xi - x20)² = 14.6, give a confidence interval with level of confidence 90% for the value of mu.

4) We now suppose we know mu is known and is equal to 2. Give a confidence interval with a confidence level of 90% for the value of sigma².

5) Same question if we do not know the value of mu.

Please see attachment for proper format.

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##### Solution Summary

The response provides the confidence interval at 90% confidence, a new sample size based on the shorter interval, how the confidence interval would be found without the variance, and when sigma and mu variables are known.

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

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