Copyright © 2012 by Nelson Education Limited. Chapter 6 Estimation Procedures 6-1.

Copyright © 2012 by Nelson Education Limited.

Chapter 6Estimation Procedures

6-1


• The logic of estimation and key terms• How to construct and interpret interval

estimates for:– Sample means– Sample proportions

• Width of confidence intervals

In this presentation you will learn about:

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• In estimation procedures, statistics calculated from random samples are used to estimate the value of population parameters.

Basic Logic of Estimation

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–If we know 71% of a random sample drawn from a city perceive lawn pesticides to be a source of environmental pollution, we can estimate the percentage of all city residents who perceive these products to be a source of environmental pollution.

Basic Logic of Estimation: An Example

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• Information from samples is used to estimate information about the population.

• Statistics are used to estimate parameters.

POPULATION

SAMPLE

PARAMETER

STATISTIC

Basic Logic of Estimation (continued)

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• Sampling Distribution is the link between sample and population.– The value of the

parameters are unknown but characteristics of the sampling distribution are defined by theorems (see Ch. 5).

POPULATION

SAMPLING DISTRIBUTION

SAMPLE

Basic Logic of Estimation (continued)

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1. A point estimate is a sample statistic used to estimate a population value.

Example: A newspaper story reports that 15% of a sample of randomly selected Canadians did not have a regular medical doctor.

Two Estimation Procedures

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2. Confidence intervals consist of a range of values.

Example: Between 12% and 18% of Canadians did not have a regular medical doctor.

Two Estimation Procedures (continued)

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• Criteria for choosing estimators:

– Bias: An estimator is unbiased if the mean of its sampling distribution is equal to the population value of interest. Sample means and proportions are unbiased estimators (Figure 6.1).

– Efficiency: Extent to which the sampling distribution is clustered around its mean. Larger samples have greater clustering around mean and thus more efficiency (Figures 6.2 and 6.3).

Bias and Efficiency

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Bias and Efficiency (continued)

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Bias and Efficiency (continued)

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• First, set the alpha, α (probability that the interval will be wrong).– Example: Setting alpha equal to 0.05, a 95% confidence level, means

the researcher is willing to be wrong 5% of the time.

• Second, find the Z score associated with alpha. – Example; If alpha is equal to 0.05, we would place half (0.025) of this

probability in the lower tail and half (0.025) in the upper tail of the distribution.

* The procedures for constructing confidence intervals for small samples (n<100) are not covered in this text.

Constructing Confidence Intervals for Means (for Large Samples)*

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Constructing Confidence Intervals for Means (for Large Samples)

(continued)

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* In Formula 6.2 σ is replaced by s. Further, n is replaced by n-1 to correct for the fact that s is a biased estimator of σ.

Constructing Confidence Intervals for Means (for Large Samples)

(continued)

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• A random sample of 178 Canadian households watches TV an average of 6 hours per day, with a standard deviation of 3 (s=3).

• With alpha set to .05, the confidence interval is:

– c.i. = 6.0 ±1.96(3/√178-1)– c.i. = 6.0 ±1.96(3/13.30)– c.i. = 6.0 ±1.96(.23)– c.i. = 6.0 ± .44

Constructing Confidence Intervals for Means: An Example

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• We can estimate that households in Canada average 6.0 ± .44 hours of TV watching each day.

• Another way to state the interval:– 5.56≤μ≤6.44

• We estimate that the population mean is greater than or equal to 5.56 and less than or equal to 6.44.


(continued)

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• This interval has a .05 (5%) chance of being wrong.–Even if the statistic is as much as ±1.96 standard

deviations from the mean of the sampling distribution the confidence interval will still include the value of μ.

–Only rarely (5 times out of 100) will the interval not include μ.


(continued)

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• Procedures:1. Set alpha.

2. Find the associated Z score.

3. Substitute values into the formula for constructing confidence intervals for sample proportions:

*The procedures for constructing confidence intervals for small samples (n<100) are not covered in this text.

Constructing Confidence Intervals for Proportions (for Large Samples)*

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where Ps = sample proportion

Z = Z score as determined by the alpha level Pu = population proportion (Pu is typically setting

at .5) = standard deviation of the sampling distribution of sample proportions

Constructing Confidence Intervals for Proportions (for Large Samples) (continued)

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• If 22% of a random sample of 764 adult Canadians smoke, what % of all adult Canadians smoke?

– c.i. = .22 ±1.96 (√.25/764)

– c.i. = .22 ±1.96 (√.00033)

– c.i. = .22 ±1.96 (.018)

– c.i. = .22 ±.04

Constructing Confidence Intervals for Proportions: An Example

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• Changing back to %’s, we can estimate that 22% ± 4% of Canadian adults smoke.

• Another way to state the interval:– 18%≤Pu≤ 26%

– We estimate the population value is greater than or equal to 18% and less than or equal to 26%.

• This interval has a .05 chance of being wrong.

Constructing Confidence Intervals for Proportions: An Example (continued)

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• In the smoking example above, can you identify the following?– Population– Sample– Statistic– Parameter

QUIZ

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• Population = All Canadian adults.

• Sample = the 764 people selected for the sample and interviewed.

• Statistic = Ps = .22 (or 22%)

• Parameter = unknown. The % of all adult Canadians who smoke.

QUIZ (continued)

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Confidence interval widens as confidence level increases:

Confidence interval narrows as sample size increases:

Controlling the Width of Confidence Intervals

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Copyright © 2012 by Nelson Education Limited. Chapter 6 Estimation Procedures 6-1.

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