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Confidence Intervals and Biased Estimators

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6. The heights of a random sample of 50 college students showed a mean of 174.5 cm and
standard deviation of 6.9 cm.

a) Construct a 98% confidence interval for the mean height of all college students.
b) What can we assert with 98% confidence about the possible size of our error if we
estimate the mean height of all college students to be 174.5 cm?

8. An electrical firm manufactures light bulbs that have a length of life that is approximately
normally distributed with a standard deviation of 40 hrs. A sample of 30 bulbs has an
average life of 780 hrs. How large a sample is needed if we wish to be 96% confident that
our sample mean will be within 10 hr of the true mean?

14. A random sample of 10 chocolate energy bars of a certain brand has, on average, 230
calories with a standard deviation of 15 calories. Construct a 99% confidence interval for
the true mean calorie content of this brand of energy bar. Assume that the distribution of
the calories is approximately normal.

22. Consider E(S'2) = E[(n-1/n)S2] = (n-1/n)*E(S2) = (n-1/n)ơ2 and S'2, the estimator of ơ2.
Analysts often use S'2

a) What is the bias of S'2?
b) Show that the bias of S'2 approaches zero as n --> infinity.

See attached file for full problem description.

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Confidence Intervals and Biased Estimators are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Solution provided by:
Changping Wang, MA
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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