62: Estimating a Population Mean: Large Samples

1
6-2 Estimating a Population Mean Large
Samples

• Objective
• Given a large (more than 30) collection of sample
  values, develop an estimate of the population
  mean.
• Assumptions being
• Sample size gt 30
• The sample is a simple random sample
• All selections have an equal chance of being
  selected.

2
Definitions

• Estimator a formula or process for using sample
  data to estimate a population parameter.
• Estimate a specific value or range of values
  used to approximate a population parameter.
• Point estimate a single value used to
  approximate a population parameter.
• (The sample mean x-bar is the best point estimate
  of the population mean.)

3
Definitions

• Estimate
• Confidence Interval the range or interval of
  values used to estimate the true value of the
  population mean.
• (56, 65), or 56 lt ? lt 65
• Alpha region the region under the probability
  curve not included in the level of confidence.
• Degree / Level of confidence the probability
  that the confidence interval actually does
  contain the population parameter. ( 1 - ? )

4
Definitions

• Confidence Area

?/2 region
5
Degree / Level of Confidence

• 3 most common are
• ? 0.1 1-0.1 0.9 90 level of confidence.
• ? 0.05 1-0.05 0.95 95 level of
  confidence.
• ? 0.01 1-0.01 0.99 99 level of confidence.
• Example
• The .95 or 95 level of confidence interval
  estimate of the population mean is (98.08,
  98.32)
• What is ? for a 95 level of confidence?

6
Interpretation (98.08, 98.32)

• Correct We are 95 confident that the interval
  (98.08, 98.32) actually does contain the true
  value of ?, the population mean.
• If we were to select many different samples of
  the same size and construct the confidence
  intervals, 95 of them would actually contain the
  value of ?.
• Incorrect There is a 95 chance that the true
  value of ? actually falls in the interval (98.08,
  98.32)

7
Confidence Intervals from 20 Different Samples
Figure 6-1
8
Definition Critical Value

• The number on the borderline (designated ? z?/2)
  separating sample statistics that are likely to
  occur from those that are unlikely to occur. The
  number z?/2 is a critical value that is a z-score
  with the property that it separates an area of
  in the right tail of the standard normal
  distribution.

9
The Critical Value
z??2
??2
??2
z??2
-z??2
z0
Found from Table A-2 (corresponds to area of 0.5
- ??2 )
Figure 6-2
10
Find the critical values corresponding to a __
degree of confidence.
11
Margin of Error (?)

• the maximum likely difference between the
  observed sample mean and the population mean.

12

• Margin of Error

13
Calculating ? when ? is unknown(most likely
scenerio)

• If n gt 30 we can replace ? in Formula 6-1 with s,
  the sample standard deviation.
• If n ? 30, the population must have a normal
  distribution, and we must know ? to use Formula
  6-1.

14
Confidence Interval (or Interval Estimate) for
Population Mean µ(Based on Large Samples n gt30)
15
Rounding rules

• When using the original set of data, round the
  confidence interval limits to one more decimal
  place than used in the original set of data.
• When the original set of dat is unknown and only
  the summary statistics are used, round the CI
  limits to the same number of decimal places used
  for the sample mean.

16
Procedure for Constructing a Confidence Interval
for µ( Based on a Large Sample n gt 30 )
17
Procedure for Constructing a Confidence Interval
for µ( Based on a Large Sample n gt 30 )

• 1. Find the critical value z??2 that corresponds
  to the desired degree of confidence.

18
Procedure for Constructing a Confidence Interval
for µ( Based on a Large Sample n gt 30 )

• 1. Find the critical value z??2 that corresponds
  to the desired degree of confidence.

2. Evaluate the margin of error ? z??2 ? /
n .
If the population standard deviation ? is
unknown,  use the value of the sample standard
deviation s  provided that n gt 30.
19
Procedure for Constructing a Confidence Interval
for µ( Based on a Large Sample n gt 30 )

• 1. Find the critical value z??2 that corresponds
  to the desired degree of confidence.

2. Evaluate the margin of error E z??2 ? /
n .
If the population standard deviation ? is
unknown,  use the value of the sample standard
deviation s  provided that n gt 30.
3. Find the values of x - E and x E.
Substitute those
values in the general format of the confidence
interval

20
Procedure for Constructing a Confidence Interval
for µ( Based on a Large Sample n gt 30 )

• 1. Find the critical value z??2 that corresponds
  to the desired degree of confidence.

2. Evaluate the margin of error E z??2 ? /
n .
If the population standard deviation ? is
unknown,  use the value of the sample standard
deviation s  provided that n gt 30.
3. Find the values of x - E and x E.
Substitute those
values in the general format of the confidence
interval
4. Round using the confidence intervals roundoff
rules.
21
Example A study found the body temperatures of
106 healthy adults. The sample mean was 98.2
degrees and the sample standard deviation was
0.62 degrees. Find the margin of error E and the
95 confidence interval.
22
Example A study found the body temperatures of
106 healthy adults. The sample mean was 98.2
degrees and the sample standard deviation was
0.62 degrees. Find the margin of error E and the
95 confidence interval.

• n 106
• x 98.2o
• s 0.62o
• ? 0.05
• ??/2 0.025
• z ?/ 2 1.96

23
Example A study found the body temperatures of
106 healthy adults. The sample mean was 98.2
degrees and the sample standard deviation was
0.62 degrees. Find the margin of error E and the
95 confidence interval.

• n 106
• x 98.20o
• s 0.62o
• ? 0.05
• ??/2 0.025
• z ?/ 2 1.96

24
Example A study found the body temperatures of
106 healthy adults. The sample mean was 98.2
degrees and the sample standard deviation was
0.62 degrees. Find the margin of error E and the
95 confidence interval.

• n 106
• x 98.20o
• s 0.62o
• ? 0.05
• ??/2 0.025
• z ?/ 2 1.96

25
Example A study found the body temperatures of
106 healthy adults. The sample mean was 98.2
degrees and the sample standard deviation was
0.62 degrees. Find the margin of error E and the
95 confidence interval.

• n 106
• x 98.20o
• s 0.62o
• ? 0.05
• ??/2 0.025
• z ?/ 2 1.96

E z ?/ 2 ? 1.96 0.62 0.12
n
106
x - E lt ? lt x E
98.20o - 0.12 lt ? lt 98.20o 0.12
26
Example A study found the body temperatures of
106 healthy adults. The sample mean was 98.2
degrees and the sample standard deviation was
0.62 degrees. Find the margin of error E and the
95 confidence interval.

• n 106
• x 98.20o
• s 0.62o
• ? 0.05
• ??/2 0.025
• z ?/ 2 1.96

E z ?/ 2 ? 1.96 0.62 0.12
n
106
x - E lt ? lt x E
98.20o - 0.12 lt ? lt 98.20o 0.12
98.08o lt ? lt 98.32o
27
Example A study found the body temperatures of
106 healthy adults. The sample mean was 98.2
degrees and the sample standard deviation was
0.62 degrees. Find the margin of error E and the
95 confidence interval.

• n 106
• x 98.20o
• s 0.62o
• ? 0.05
• ??/2 0.025
• z ?/ 2 1.96

E z ?/ 2 ? 1.96 0.62 0.12
n
106
x - E lt ? lt x E
98.08o lt ? lt 98.32o
Based on the sample provided, the confidence
interval for the population mean is 98.08o lt ?
lt 98.32o. If we were to select many different
samples of the same size, 95 of the confidence
intervals would actually contain the population
mean ?.
28
Example The drive through service times were
recorded for 52 randomly selected customers at a
Burger King Restaurant. Those times had a mean
of 181.3 sec and a s.d. of 82.2 sec. Construct a
95 confidence interval estimate of the
population mean.

• n 52
• x 181.3
• s 82.2
• ? 0.05
• ??/2 0.025
• z ?/ 2 1.96

29
Example The drive through service times were
recorded for 52 randomly selected customers at a
Burger King Restaurant. Those times had a mean
of 181.3 sec and a s.d. of 82.2 sec. Construct a
95 confidence interval estimate of the
population mean. n 52 x 181.3 s 82.2 ?
0.05 ??/2 0.025 z ?/ 2 1.96
E z ?/ 2 ? 1.96 82.2 22.3
x - E lt ? lt x E
159.0 lt ? lt 203.6
Based on the sample provided, the confidence
interval for the population mean is 159.0 lt ? lt
203.6. If we were to select many different
samples of the same size, 95 of the confidence
intervals would actually contain the population
mean ?.
30
Finding the Point Estimate and E from a
Confidence Interval
Point estimate of µ x (upper confidence
interval limit) (lower confidence interval
limit) 2
31
Finding the Point Estimate and E from a
Confidence Interval
Point estimate of µ x (upper confidence
interval limit) (lower confidence interval
limit) 2
Margin of Error E (upper confidence interval
limit) - (lower confidence interval limit)
2
32
Finding the Point Estimate and E from a
Confidence Interval

• Find the point estimate of the mean and the
  margin of error for the following confidence
  intervals
• (254.6, 305.1)
• ? E

33
Finding the Point Estimate and E from a
Confidence Interval

• Find the point estimate of the mean and the
  margin of error for the following confidence
  intervals
• (254.6, 305.1)
• ? E

• 28.3 lt ? lt 40.4
• ? E