Estimation Procedures: Constructing Confidence Intervals Using the Sampling Distribution, Slides of Statistics for Psychologists

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Healey Chapter 7

Estimation Procedures

Using the Sampling Distribution

to Construct Confidence

Intervals

Outline:

• The logic of estimation
• How to construct and interpret confidence

interval estimates for:

• Sample means
• Sample Proportions

Logic (cont.)

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

POPULATION

SAMPLE

PARAMETER

STATISTIC

Logic (cont.)

• Sampling Distribution is the link between sample and population.
• The value of the parameters is unknown but characteristics of the Sampling Distribution are defined by theorems.

POPULATION

SAMPLING DISTRIBUTION

SAMPLE

Bias and Efficiency

• Bias:
  • An estimator of a mean (or a proportion) is unbiased if the mean of its sampling distribution is equal to the population mean.
• Efficiency:
  • The smaller the standard error (S.D. of the sampling distribution,) the more the samples are clustered about the mean of the S.D.
  • This is known as efficiency.

Sample Size and Efficiency

• Standard error of sampling distribution:
• =
• In looking at the formula, we can see that as sample size N increases, the standard error ( ) will decrease. The larger N is, the more efficient the estimate will be. A larger sample size means that the estimate is closer to the real population mean.

N − 1

S

Confidence Levels (cont.) When α = .05…

…then .025 of the area is distributed on either side (C )

The .95 in the middle section is our confidence level.

The cut-off between our confidence level and +/- .025 is represented by a Z-value of +/- 1.96.

c c

Z-values for Various Alpha Levels

Confidence Level α α/2 Z-score

( Note: Z-scores are found in Appendix A using the area for α/2)

Example: Confidence Intervals For Means

• Question:
• For a random sample of 178 Canadian

households, average television viewing time was 6 hours/day with s = 3. What would be your estimate of the population mean viewing time, at the 95% confidence level (Alpha (α) = .05)

Example: Confidence Intervals For

• Z-score for 95% confidence level (Means α+.05) is +/-1.
• Substitute all information into formula and solve:

c.i. =



 

 

 

 −

Χ ±Ζ N 1

s

Example (cont.)

• In other words:
• Even if the statistic is as much as ±1.

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.

Confidence Intervals For Proportions

• Procedure:
  • Set alpha = .05.
  • Find the associated Z score.
  • Substitute the sample information into formula:

c.i. =

Note: Ρ s = sample proportion Ρ u (when population proportion is not known,) is set to.

( ) Ν

Ρ −Ρ Ρ ±Ζ

u u s

1

Example for Proportions (cont.)

c.i. =

( ) Ν

Ρ −Ρ Ρ ± Ζ

u u s

1

Confidence Intervals For Proportions

• Changing back to %, we estimate that 42% ± 4% of the city residents vote Liberal.
• Another way to state the interval:

38% ≤ Pu ≤ 46%

Interpretation : We estimate that the population value is greater than or equal to 38% and less than or equal to 46% for city residents who vote Liberal.

(This interval has a .05 chance of being wrong.)