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# Probability Problem Set

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Problem 3
a. A random sample of 25 was drawn from a normal distribution with a standard deviation of 5. The sample mean is 80. Determine the 95% confidence interval estimate of the population mean.
b. Repeat part (a) with a sample size of 100.
c. Repeat part (a) with a sample size of 400.
d. Describe what happens to the confidence interval estimate when the sample size increases.

Problem 4
a. A statistics practitioner randomly sampled 100 observations from a population with a standard deviation of 5 and found that is 10. Estimate the population mean with 90% confidence.
b. Repeat part (a) with a sample size of 25.
c. Repeat part (a) with a sample size of 10.
d. Describe what happens to the confidence interval estimate when the sample size decreases.

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3. a) The critical value for 95% confidence interval is 1.96.
a.margin of error=1.96*5/sqrt(25)=1.96
upper limit: 80+1.96=81.96
lower limit: 80-1.96=78.04
Therefore, 95% confidence interval is [78.04, 81.96].

b) margin of error=1.96*5/sqrt(100)=0.98
upper limit: 80+0.98=80.98
lower limit: ...

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