Section 6.1 Confidence Intervals for the Mean (Large Samples) Larson/Farber 4th ed.

Section 6.1

Confidence Intervals for the Mean (Large Samples)

Larson/Farber 4th ed

Section 6.1 Objectives

• Find a point estimate and a margin of error• Construct and interpret confidence intervals for the

population mean• Determine the minimum sample size required when

estimating μ


Point Estimate for Population μ

Point Estimate• A single value estimate for a population parameter• Most unbiased point estimate of the population mean

μ is the sample mean

Estimate Population Parameter…

with Sample Statistic

Mean: μ


Example: Point Estimate for Population μ

A random sample of annual precipitation from Nome, Alaska. Find a point estimate of the population mean, μ.

18.31 12.29 14.30 19.25 19.87 20.14 7.39 14.92 19.76 24.3820.09 13.05 17.10 22.15 9.93 14.17 19.06 24.25 14.97 15.4614.93 13.67 17.13 16.27 9.08 20.80 10.44 15.23 20.66 17.4922.06 13.43 17.62

Solution: Point Estimate for Population μ

The sample mean of the data is

Your point estimate for the mean precipitation for Nome, Alaska is 16.66 inches.


Interval Estimate

Interval estimate • An interval, or range of values, used to estimate a

population parameter.

How confident do we want to be that the interval estimate contains the population mean μ?


Point estimate

• 16.66

( )Interval estimate

Level of Confidence

Level of confidence c • The probability that the interval estimate contains the

population parameter.

zz = 0-zc zc

Critical values

½(1 – c) ½(1 – c)

c is the area under the standard normal curve between the critical values.

The remaining area in the tails is 1 – c .

c

Use the Standard Normal Table to find the corresponding z-scores.


−zc

Level of Confidence

• If the level of confidence is 90%, this means that we are 90% confident that the interval contains the population mean μ.

zz = 0 zc

The corresponding z-scores are +1.645.

c = 0.90

½(1 – c) = 0.05½(1 – c) = 0.05


Sampling Error

Sampling error • The difference between the point estimate and the

actual population parameter value.• For μ:

the sampling error is the difference – μ μ is generally unknown varies from sample to sample


Margin of Error

Margin of error • The greatest possible distance between the point

estimate and the value of the parameter it is estimating for a given level of confidence, c.

When n ≥ 30, the sample standard deviation, s, can be used for σ.

• Denoted by E.

• Sometimes called the maximum error of estimate or error tolerance.

Example: Finding the Margin of Error

Use the annual precipitation data and a 95% confidence level to find the margin of error for the mean number inches of precipitation per year.

n = 33

Solution: Finding the Margin of Error

• First find the critical values

zzcz = 0

0.95

0.0250.025

-zc = -1.96

95% of the area under the standard normal curve falls within 1.96 standard deviations of the mean. (You can approximate the distribution of the sample means with a normal curve by the Central Limit Theorem, because n ≥ 30.)


Solution: Finding the Margin of Error

You don’t know σ, but since n ≥ 30, you can use s in place of σ.

You are 95% confident that the margin of error for the population mean is about 1.4 inches of precipitation.


Confidence Intervals for the Population Mean

A c-confidence interval for the population mean μ


The probability that the confidence interval contains μ is c.

Constructing Confidence Intervals for μ

Finding a Confidence Interval for a Population Mean (n ≥ 30 or σ known with a normally distributed population)

In Words In Symbols

1. Find the sample statistics n and .

2. Specify σ, if known. Otherwise, if n ≥ 30, find the sample standard deviation s and use it as an estimate for σ.


Constructing Confidence Intervals for μ

3. Find the critical value zc that corresponds to the given level of confidence.

4. Find the margin of error E.

5. Find the left and right endpoints and form the confidence interval.

Use the Standard Normal Table.

Left endpoint: Right endpoint: Interval:


In Words In Symbols

Example: Constructing a Confidence Interval

Construct a 95% confidence interval for the mean annual precipitation for Nome Alaska.Solution: Recall the mean was 16.66 and E = 1.44

15.2 < μ

Left Endpoint:Right Endpoint:

< 18.1

Example: Constructing a Confidence Interval

( )• 16.6615.22 18.10

With 95% confidence, you can say that the mean annual precipitation for Nome, Alaska is between 15.22 and 18.10 inches.

Point estimate

Interpreting the Results

• μ is a fixed number. It is either in the confidence interval or not.

• Incorrect: “There is a 95% probability that the actual mean is in the interval (15.22,18.10).”

• Correct: “If a large number of samples is collected and a confidence interval is created for each sample, approximately 95% of these intervals will contain μ.


Interpreting the Results

The horizontal segments represent 90% confidence intervals for different samples of the same size.In the long run, 9 of every 10 such intervals will contain μ.


μ

Sample Size

• Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate the population mean μ is

• If σ is unknown, you can estimate it using s provided you have a preliminary sample with at least 30 members.

Example: Sample Size

You want to estimate the mean annual rain fall for Nome, Alaska. How many years must be included in the sample if you want to be 99% confident that the sample mean is within one standard error of the population mean?

−zc

Solution: Sample Size

• First find the critical values

zc = 2.575

zz = 0 zc

0.99

0.0050.005

Solution: Sample Size

zc = 2.575 σ ≈ s = 4.2 E = 1

You should include at least 117 years in your sample.

Section 6.1 Summary

• Found a point estimate and a margin of error• Constructed and interpreted confidence intervals for

the population mean• Determined the minimum sample size required when

estimating μ


Section 6.1 Confidence Intervals for the Mean (Large Samples) Larson/Farber 4th ed.

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Transcript of Section 6.1 Confidence Intervals for the Mean (Large Samples) Larson/Farber 4th ed.