Download Confidence Intervals for the Mean: Known

Math 221: Confidence Intervals for the Mean: σ known
S. K. Hyde
Chapter 14 (Moore, 5th Ed.)
The health of the bear population in Yellowstone National Park is monitored by periodic measurements
taken from anesthetized bears. A sample of 54 bears has a mean weight of 182.9 lbs. Assume that σ is
known to be 121.8 lb.
1. Find a 99% confidence interval for the mean of the population of all such bear weights.
2. What aspect of this problem is not realistic?
Sample size example:
Nielsen Media Research wants to find out the mean amount of time (in min) a full time college student
spends watching TV.
1. Find the sample size necessary to estimate the mean within 15 minutes assuming a 96%
confidence level (assume a pilot study showed σ = 112.2).
Confidence Intervals for the Mean: σ Known, page 2
Solution
1. Find a 99% confidence interval for the mean of the population of all such bear weights.
√
= 16.57488. The margin of error is
First, z ∗ = 2.576. Next, SEx̄ = √σn = 121.8
54
∗
m = z SEx̄ = 42.69689. So a 99% confidence interval for the mean of the bear weights is
x̄ ± m
x̄ ± z ∗ SEx̄
182.9 ± 42.69689
(140.2031, 225.5969).
Note: When using a TI-83 or TI-84 calculator, select STAT −→ TESTS −→ ZInterval , and
enter the appropriate data or statistics.
2. What aspect of this problem is not realistic?
It is not realistic to know the population standard deviation for the weights of the bears.
Sample size example:
1. Find the sample size necessary to estimate the mean within 15 minutes assuming a 96%
confidence level (assume a pilot study showed σ = 112.2).
So
z∗σ
n=
m
2
(2.054)(112.2)
=
15
2
= 236.0500
Since n must be an integer, we must round n up each time. So, in this case
n = 237.