Download Lesson 8.1 Estimation µ when σ is Known Notes

Statistics
Lesson 8.1 Estimation µ when σ is Known Notes
Page 1 of 4
Assumptions about the random variable x
1. We have a simple random sample of size n drawn from a population of x
values.
2. The value of σ, the population standard deviation of x, is known
3. If the x distribution is normal, then our methods work for any sample size n.
4. If x has an unknown distribution then we require a sample size n ≥ 30.
However, if the x distribution is distinctly skewed and definitely not moundshaped, or bell shaped, then a sample size of 50 or higher may be necessary.
Point Estimate:
 A point estimate of a population parameter is an estimate of the parameter
using a single number.
o x is the point estimate for µ
 When using x as a point estimate for µ, the margin of error is the magnitude
of x   or x   .
Confidence level (c)
 The reliability of an estimate.
 For a confidence level c, the critical value zc is the number such that the area
under the standard normal curve between – zc and zc equals c.
 The area under the normal curve from – zc and zc, is the probability that the
standardized normal variable z lies in the interval. This means that
P( zc  z  zc ) .
Example 1:
Find the critical value zc for c = 0.90.
Example 2:
Find P( z0.99  z  z0.99 )  0.99 .
Statistics
Lesson 8.1 Estimation µ when σ is Known Notes
Page 2 of 4
c Confidence Interval for µ
 A c Confidence Interval for µ is an interval computed from sample data in such a
way that c is the probability of generating an interval containing the actual value
of µ. In other words, c is the proportion of confidence intervals, based on
random samples of size n that actually contain µ.
How to find a confidence interval for µ when σ is known…
xE  xE
where x = sample mean of a simple random sample
E  zc

n
c = confidence level (0 < c < 1)
zc = critical value for confidence level c based on the standard normal
distribution.
Example 3:
Julia enjoys jogging. She has been jogging over a period of several years during which
time her physical condition has remained constantly good. Usually, she jogs 2 miles
per day. The standard deviation of her times is σ = 1.80 minutes. During the past
year, Julia has recorded her times to run 2 miles. She has a random sample of 90 of
these times. For these 90 times, the mean was x = 15.60 minutes. Let µ be the mean
jogging time for the entire distribution of Julia’s 2 mile running times (taken over the
past year.) Find a 0.95 confidence interval for µ. What does it mean?
Statistics
Lesson 8.1 Estimation µ when σ is Known Notes
Page 3 of 4
Example 4:
Walter usually meets Julia at the track. He prefers to jog 3 miles. From long
experience, he knows that σ = 2.40 minutes for his jogging times. For a random
sample of 90 sessions, Walter’s mean time was x = 22.50 minutes. Let µ be the mean
jogging time for the entire distribution of Walter’s 3 mile running times over the past
several years. Find a 0.99 confidence interval for µ. What does it mean?
Example 5:
Suppose that the standard deviation of all high school seniors’ SAT scores in a certain
year was σ = 150. A random sample of 100 scores yielded the sample mean x =
1010. Let µ be the mean of all SAT scores in that year. Find a 0.99 confidence
interval for µ. Round your answers to integers.
Statistics
Lesson 8.1 Estimation µ when σ is Known Notes
Page 4 of 4
Example 6:
Use the given information in Example 5.
a. What is the value of z0.95 ?
b. Is the x distribution approximately normal?
c. What is the value E when c = 0.95? Round your answer to an integer.
d. What are the endpoints for a 0.95 confidence interval for µ? Round your
answers to integers.
e. Interpret the confidence interval.