Download 7.3 Estimating a Population Mean

7.3 – Estimating a Population Mean when
Standard Deviation is Known:
Objectives:
1.
2.
3.
4.
5.
Find critical values.
Determine margin of error.
Find point estimate for the population mean.
Construct confidence interval for mean when standard deviation is known.
Determine the sample size required to estimate the mean.
Overview:
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, σ. Here are
three key concepts that should be learned in this section:
1. We should know that the sample mean is the best point estimate of the population mean µ.
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.
Sample Mean as a Point Estimate:
1. For all populations, the sample mean x is an unbiased estimator of the population mean µ,
meaning that the distribution of sample means tends to center about the value of the population
mean µ.
2. The sample mean x is the best point estimate of the population mean µ.
3. For many populations, the distribution of sample means x tends to be more consistent (with less
variation) than the distributions of other sample statistics.
4. However, just as with sample proportions, we do not know just how good an estimate the sample
mean provides, therefore more information can be provided by using confidence intervals which
provide a range of values rather than just a single point estimate.
Margin of Error:
When data from a simple random sample are used to estimate a population mean, the margin of error,
denoted by E, is the maximum likely difference (with probability 1 – α, such as 0.95) between the
observed mean and the true value of the population mean.
The margin of error E is also called the maximum error of the estimate and can be found by multiplying
the critical value and the standard deviation divided by the square root of the sample size:
E = zα 2 ⋅
σ
n
Constructing a Confidence Interval for Estimating a Population Mean
(with σ Known):
Notation:
µ = population mean
σ = population standard deviation
x = sample mean
n = number of sample values
E = margin of error
zα/2 = z score separating an area of a/2 in the right tail of the standard normal distribution
Requirements:
1. The sample is a simple random sample.
2. The value of the population standard deviation σ is known.
3. Either or both of these conditions is satisfied: The population is normally distributed or n > 30.
Confidence Interval:
The confidence interval may be written as:
x−E<µ<x+E
where
E = zα 2 ⋅
σ
n
The confidence interval is often expressed in the following equivalent forms:
x±E
or
( x − E, x + E)
Procedure for Constructing Confidence Intervals:
1.
2.
3.
4.
Verify that requirements are met.
Find the critical value for the confidence level.
Find the margin of error.
Determine confidence interval.
Interpreting a Confidence Interval:
We must be careful to interpret confidence intervals correctly. There is a correct interpretation and many
different and creative incorrect interpretations of the confidence interval 164.9 < µ < 180.6.
Correct Interpretation:
“We are 95% confident that the interval from 164.9 to 180.6 actually does contain the true value of the
population mean.” This means that if we were to select many different samples of the same size and
construct the corresponding confidence intervals, 95% of them would actually contain the value of the
population mean.
Incorrect Interpretation:
There is a 95% chance that the true value of the population mean will fall between 164.9 and 180.6. It
would also be incorrect to say that “95% of sample means fall between 164.9 and 180.6
Example: A random sample of 105 light bulbs had a mean life of x = 441 hours with a standard
deviation of σ = 40 Construct a 90% confidence interval for the mean life, µ, of all light bulbs of this
type.
Solution: Assume requirements are met. The z-score corresponding to a 90% confidence level is 1.645.
The margin of error is:
E = zα 2 ⋅
σ
n
 40 
E = 1.645

105


E = 1.645(3.904)
E = 6.421
The confidence interval x − E < µ < x + E is:
441 − 6.421 < µ < 441 + 6.421
434.579 < µ < 447.421
435 < µ < 447
We can be 90% confident that the actual mean life of a light bulb will fall between 435 and 447.
Example: A random sample of 130 full-grown lobsters had a mean weight of 21 ounces and a standard
deviation of 3.0 ounces. Construct a 98% confidence interval for the population mean µ.
Solution: Assume requirements are met. The z-score corresponding to a 98% confidence level is 2.33.
The margin of error is:
E = zα 2 ⋅
σ
n
 3 
E = 2.33

130


E = 2.33(0.263)
E = 0.613
The confidence interval x − E < µ < x + E is:
21 − 0.613 < µ < 21 + 0.613
20.387 < µ < 21.613
20 < µ < 22
We can be 98% confident that the actual mean weight of a lobster will fall between 20 and 22.
Example: A group of 66 randomly selected students have a mean score of 22.4 with a standard
deviation of 2.8 on a placement test. What is the 90% confidence interval for the mean score, µ, of all
students taking the test?
Solution: Assume requirements are met. The z-score corresponding to a 90% confidence level is 1.645.
The margin of error is:
E = zα 2 ⋅
σ
n
 2.8 
E = 1.645

66


E = 1.645(0.345)
E = 0.567
The confidence interval x − E < µ < x + E is:
22.4 − 0.567 < µ < 22.4 + 0.567
21.833 < µ < 22.967
22 < µ < 23
We can be 90% confident that the actual mean score will fall between 22 and 23.
Determining Sample Size:
Suppose we want to collect sample data in order to estimate some population mean. The question is
how many sample items must be obtained? If we solve the formula for margin of error, we obtain the
following formula:
 zα 2 ⋅ σ 
n=

E


2
This formula shows that the sample size does not depend on the size of the population, but rather, on the
desired confidence level and margin of error and sometimes p̂ .
Round Off Rule for Sample Size:
If the computed sample size is not a whole number, then round the value up to the next whole number.
Example: How many business students must be randomly selected to estimate the mean monthly
earnings of business students at one college? We want 95% confidence that the sample mean is within $
128 of the population mean, and the population standard deviation is known to be $536.
Solution: Critical value = 1.96; Margin of error = 128, and standard deviation = 536.
 zα 2 ⋅ σ 
n=

 E 
2
1.96(536) 
n=
 128 
n = 67.36
2
n ≈ 67
The sample should consist of at least 67 students.
Example: How many weeks of data must be randomly sampled to estimate the mean weekly sales of a
new line of athletic footwear? We want 99% confidence that the sample mean is within $200 of the
population mean, and the population standard deviation is known to be $1100.
Solution: Critical value = 2.575; Margin of error = 200, and standard deviation = 1100.
 zα 2 ⋅ σ 
n=

 E 
2
 2.575(1100) 
n=

200

n = 200.576
n ≈ 201
The sample should consist of at least 201 weeks of data.
2
Example: How many women must be randomly selected to estimate the mean weight of women in one
age group? We want 90% confidence that the sample mean is within 3.7 lb of the population mean, and
the population standard deviation is known to be 28 lb.
Solution: Critical value = 1.645; Margin of error = 3.7, and standard deviation = 28.
 zα 2 ⋅ σ 
n=

 E 
2
1.645(28) 
n=
 3.7 
n = 154.969
n ≈ 155
2
The sample should consist of at least 155 women.
Finding the Sample Size n when σ is Unknown:
In reality, the standard deviation of the population is usually unknown. When determining a required
sample size there are ways to work around the problem of not knowing the value of the standard
deviation.
1. Use the range rule of thumb.
2. Start the sample collection process without knowing σ and, using the first several values,
calculate the sample standard deviation s and use it in place of σ . The estimated value of σ can
then be improved as more sample data are obtained, and the sample size can be refined
accordingly.
3. Estimate the value of σ by using the results of some other study that was done earlier.