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Confidence Intervals and Biased Estimators

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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