Transcript Confidence Interval for Estimating a Population Mean

Section 7.3
Estimating a
Population Mean:
 Known
Learning Targets:
•
•
This section presents methods for estimating a population
mean.
In addition to knowing the values of the sample data or
statistics, we must also know the value of the population
standard deviation, .
There are three key concepts that should be learned in this
section….
Learning Targets (continued):
1. We should know that the sample mean x is the best point estimate of
the population mean .
2. We should learn how to use sample data to construct a confidence
interval for estimating the value of a population mean, and we should
know how to interpret such confidence intervals.
3. We should develop the ability to determine the sample size necessary to
estimate a population mean.
Point Estimate of the Population Mean
The sample mean x is the best point estimate of the population
mean µ.
Confidence Interval for Estimating a Population Mean
(with  Known)
 = population mean
 = population standard deviation
x = sample mean
n = number of sample values
E = margin of error
z/2 = (z*) = z score separating an area of a/2 in the
right tail of the standard normal distribution
Margin of Error for Means (with σ Known)
E  z 2 

n
Confidence Interval for Estimating a
Population Mean (with  Known)
xExE
xE
 x  E, x  E 
The two values x  E and x  E are called
confidence interval limits.
Requirements for constructing a confidence interval for
estimating a population mean with  known.
1. The sample is a simple random sample.
2. The value of the population standard deviation  is known.
3. Either or both of these conditions is satisfied: The population
is normally distributed or n > 30.
Example 1:Find the margin of error and CI if the necessary requirements
are satisfied. If the requirements are not satisfied, state why.
a) The amounts of rainfall for a simple random sample of Saturdays in Boston:
99% confidence, n = 12, x = 0.133 in.,  is known to be 0.212 in., and the
population is known to have daily rainfall amounts with a distribution that is
far from normal.
b) The braking distances of a simple random sample of trucks: 96% confidence;
n = 42, x = 147 feet, and  is known to be 9 feet.
Warm-Up/Example 2 (Quick Review from 7.2): Find the indicated critical value zα/2
a) Find the critical value zα/2 (z*) that corresponds to a 95% confidence level.
b) Find the critical value zα/2 (z*) that corresponds to a 99% confidence level.
c) Find zα/2 (z*) for α = 0.10.
d) Find zα/2 (z*) for α = 0.06.
Example 3: People have died in boat and aircraft accidents because an obsolete estimate
of the mean weight of men was used. In recent decades, the mean weight of men has
increased considerably, so we need to update our estimate of that mean so that boats,
aircraft, elevators, and other such devices do not become dangerously overloaded. Using
the weights of men from Data Set 1 in Appendix B, we obtain these sample statistics for
the simple random sample: n = 40 and
x = 172.55 lb. Research from several other sources suggests that the population of weights
of men has a standard deviation given by  = 26 lb.
a) Find the best point estimate of the mean weight of the population of all men.
b) Construct a 95% confidence interval estimate of the mean weight of all men.
Calculator Setup
1st: On your graphing calculator, go to
STAT  TESTS
2nd: Choose 7: ZInterval
There will be a choice between Data and Stats, we’re
going to choose “Stats”
For Example…
**Note: We would choose
Data if we were given a list of
data values and needed to
calculate the mean and
standard deviation.
Example 4: Find the margin of error and confidence intervals for the following
scenarios:
a) Weights of fish: 95% confidence; n = 53,x = 1.74 lb., σ = 3 lb.
b) High school students’ annual earnings: 94% confidence; n = 107, x = $1,124,
σ = $651.
x
Example 5: Using the simple random sample of weights of women from Data
Set 1 in Appendix B, we obtain these sample statistics: n = 40 and = 146.22 lb.
Research from other sources suggests that the population of weights of women
have a standard deviation given by σ = 30.86 lb.
a) Find the best point estimate of the mean weight of all women.
b) Find a 95% confidence interval estimate of the mean weight of all women.
Example 7: A simple random sample of 125 SAT scores has a mean of 1522. Assume that
SAT scores have a standard deviation of 333.
a) Construct a 95% confidence interval estimate of the mean SAT score.
b) Construct a 99% confidence interval estimate of the mean SAT score.
c) Which of the preceding confidence intervals is wider?
Finding a Sample Size for Estimating a
Population Mean
 = population mean
σ = population standard deviation
x = sample mean
E = desired margin of error
z/2 = (z*) = z score separating an area of a/2 in the
right tail of the standard normal distribution
  z *   
n

 E 
2
Round-Off Rule for Sample Size n
If the computed sample size n is not a whole
number, round the value of n up to the next
larger whole number. NEVER ROUND
DOWN!
Example 8: Assume that we want to estimate the mean IQ score for the
population of statistics students. How many statistics students must be
randomly selected for IQ tests if we want 95% confidence that the sample
mean is within 3 IQ points of the population mean? From last chapter, we
know that the standard deviation of IQ scores is 15.
Example 9: Use the given information to find the minimum sample size required
to estimate an unknown population mean µ.
a) Margin of error: $242, confidence level: 80%, σ = $754.
b) Margin of error: 1.84 lb., confidence level: 92%, σ = 12.41 lb.
Example 10: How many men must be randomly selected to estimate the mean
height of men in one age group? We want 96% confidence that the sample
mean is within 1.34 in. of the population mean, and the population standard
deviation is known to be 2.5 in.
Example 11: Refer to the TI-83/84 Plus calculator display of a 90%
confidence interval. The sample display results from using a simple random
sample of the weights of kittens at age 3 months.
a) Identify the value of the point estimate
of the population mean µ.
b) Express the confidence interval in the format of
x  E    x  E.
c) Express the confidence interval in the format of x  E.