Solution to Extra on HW 8

• Extra #1:   It is fairly well established that heights of adult women in the U.S. are normally distributed with a standard deviation of 3 inches.  A random sample of 10 women are selected and there heights are recorded as follows:

    63.6, 65.2, 62.2, 71.1, 65.8, 65.1, 71.0, 64.8, 63.3, 64.9

The five number summary is
(62.20, 63.90, 65.00, 65.65, 71.10)  and the average of the sample is 65.70.

a)Is the standard deviation of the population known or unknown in this problem?
b) Calculate the standard deviation of the sample.  How does this compare to the population standard deviation?
c) Is the mean of the population known or unknown?
d) Find a 95% confidence interval using a z-statistic as a multiplier.  (Why use the z-statistic?)
e) Find a 99% confidence interval.
f) Pretend that you were not told that the population standard deviation is 3".  Use your estimate in (b) to re-calculate the 95% confidence interval.  This time, rather than a z-statistic, use the appropriate t-statistic as a multipler.

a) It is known:  sigma = 3
b) s = 3.01
c) the mean is unknown -- we want to estimate it.  Note, the mean of the *sample* is known and equals 65.7.  But we don't know the mean of the population.
d) Use z = 1.96 for a 95% CI:  65.70 +/- 1.96*(3/sqrt(10))
(63.8, 67.6)
e) use z = 2.58  65.7 +/- 2.448   ( 63.2, 68.1)
f) Use t = 2.262 (there are n-1=10-1=9 degrees of freedom)
65.7 +/- 2.15  (63.5, 67.9)