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### Statistics

Chapter 7: Inferences Based on a Single Sample: Estimation with Confidence Interval

Where We’ve Been

- Populations are characterized by numerical measures called parameters
- Decisions about population parameters are based on sample statistics
- Inferences involve uncertainty reflected in the sampling distribution of the statistic

McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

Where We’re Going

- Estimate a parameter based on a large sample
- Use the sampling distribution of the statistic to form a confidence interval for the parameter
- Select the proper sample size when estimating a parameter

McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.1: Identifying the Target Parameter

- The unknown population parameter that we are interested in estimating is called the target parameter.

McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7.2: Large-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used to estimate the population parameter.

7.2: Large-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- Suppose a sample of 225 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours.
- What can we conclude about all college students’ television time?

7.2: Large-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- Assuming a normal distribution for television hours, we can be 95%* sure that

*In the standard normal distribution, exactly 95% of the area under the curve is in the interval

-1.96 … +1.96

7.2: Large-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- An interval estimator orconfidence intervalis a formula that tell us how to use sample data to calculate an interval that estimates a population parameter.

7.2: Large-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter.
- The confidence level is the confidence coefficient expressed as a percentage.

(90%, 95% and 99% are very commonly used.)

95% sure

7.2: Large-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- The area outside the confidence interval is called

95 % sure

So we are left with (1 – 95)% = 5% = uncertainty about µ

7.2: Large-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- Large-Sample (1-)% Confidence Interval for µ
- If is unknown and n is large, the confidence interval becomes

7.2: Large-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- If n is large, the sampling distribution of the sample
- mean is normal, and s is a good estimate of

7.3: Small-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

Large Sample

Small Sample

- Sampling Distribution on is normal
- Known or large n

Standard Normal (z)Distribution

- Sampling Distribution on is unknown
- Unknown and small n

Student’s t Distribution (with n-1 degrees of freedom)

7.3: Small-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

Large Sample

Small Sample

7.3: Small-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

* If not, see Chapter 14

7.3: Small-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- Suppose a sample of 25 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours.
- What can we conclude about all college students’ television time?

7.3: Small-Sample Confidence Interval for a Population MeanMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- Assuming a normal distribution for television hours, we can be 95% sure that

7.4: Large-Sample Confidence Interval for a Population ProportionMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- Sampling distribution of
- The mean of the sampling distribution is p, the population proportion.
- The standard deviation of the sampling distribution is

where

- For large samples the sampling distribution is approximately normal. Large is defined as

7.4: Large-Sample Confidence Interval for a Population ProportionMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- Sampling distribution of

7.4: Large-Sample Confidence Interval for a Population ProportionMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

We can be 100(1-)% confident that

where and

7.4: Large-Sample Confidence Interval for a Population ProportionMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

A nationwide poll of nearly 1,500 people … conducted by the syndicated cable television show Dateline: USA found that more than 70 percent of those surveyed believe there is intelligent life in the universe, perhaps even in our own Milky Way Galaxy.

What proportion of the entire

population agree, at the 95%

confidence level?

7.4: Large-Sample Confidence Interval for a Population ProportionMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- If p is close to 0 or 1, Wilson’s adjustment for estimating p yields better results

where

7.4: Large-Sample Confidence Interval for a Population ProportionMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

Suppose in a particular year the percentage of firms declaring bankruptcy that had shown profits the previous year is .002. If 100 firms are sampled and one had declared bankruptcy, what is the 95% CI on the proportion of profitable firms that will tank the next year?

7.5: Determining the Sample SizeMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- To be within a certain sampling error (SE) of µ with a level of confidence equal to

100(1-)%, we can solve

for n:

7.5: Determining the Sample SizeMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- The value of will almost always be unknown, so we need an estimate:
- s from a previous sample
- approximate the range, R, and use R/4
- Round the calculated value of nupwards to be sure you don’t have too small a sample.

7.5: Determining the Sample SizeMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test?

7.5: Determining the Sample SizeMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test?

7.5: Determining the Sample SizeMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

To estimate p, use

the sample proportion

from a prior study, or

use p = .5.

Round the value of n

upward to ensure the

sample size is large

enough to produce the

required level of

confidence.

- For a confidence interval on the population proportion, p, we can solve

for n:

7.5: Determining the Sample SizeMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

- How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within .01 with 90% confidence, if an initial estimate of .10 is used for p?

7.5: Determining the Sample SizeMcClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within .01 with 90% confidence, if an initial estimate of .10 is used for p?

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