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Statistical Inference: Confidence Intervals for Population Parameters, Study notes of Statistics

James Madison University (JMU)Statistics

Point and interval estimates of population parameters, focusing on confidence intervals for population proportions and means. It covers the definition, construction, and properties of confidence intervals, as well as the effects of confidence level and sample size. The document also includes examples and formulas for calculating confidence intervals.

Typology: Study notes

Pre 2010

Uploaded on 02/13/2009

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Chapter 7
Statistical Inference:
Confidence Intervals
7.1 What are Point and Interval Estimates of
Population Parameters?
A. Point Estimate of a Population Parameter
•Definition: A single number that is our “best
guess” for the unknown population parameter
•Example: When sampling from a categorical (S/F)
population, the sample proportion of S’s is a point
estimate of the population proportion p
•Example: When sampling from a quantitative
population, the sample mean is a point estimate
of a population mean
•Properties of Good Point Estimators
–The estimator is unbiased: the mean of the
sampling distribution is equal to the parame-
ter being estimated
–The estimator has a small standard error
compared to other estimators.
1
pf3
pf4
pf5

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Chapter 7

Statistical Inference:

Confidence Intervals

7.1 What are Point and Interval Estimates of Population Parameters?

A. Point Estimate of a Population Parameter

  • Definition: A single number that is our “best guess” for the unknown population parameter
  • Example: When sampling from a categorical (S/F) population, the sample proportion of S’s is a point estimate of the population proportion p
  • Example: When sampling from a quantitative population, the sample mean is a point estimate of a population mean
  • Properties of Good Point Estimators
    • The estimator is unbiased: the mean of the sampling distribution is equal to the parame- ter being estimated
    • The estimator has a small standard error compared to other estimators. 1

B. (Confidence) Interval Estimate of a Population Pa- rameter

  • Definition: A interval of numbers within which the parameter is believed to fall with a certain amount of confidence.
  • Margin of Error = multiple of standard error of point estimate, used to indicate how accurate point estimate is likely be in estimating parame- ter.
  • Left Endpoint = Point estimate - margin of error Right Endpoint = Point estimate + margin of error
  • Confidence Level of Interval Estimate = proba- bility that method produces an interval that con- tains parameter. Usually chosen close to 1, such as 0.95 or 0.90.

7.2 How Can We Construct a Confidence Inter- val to Estimate a Population Proportion?

A. Finding the 95% Confidence Interval for a Popula- tion Proportion

  • Population proportion is symbolized by p
  • Sample proportion is symbolized by ˆp p ˆ is point estimate of p
  • 90% confidence level - use ˆp ± 1 .645(se) 0 .90 is middle area under standard normal curve between − 1 .645 and 1. 645 1 .645(se) is the 90% margin of error for ˆp (or the confidence interval)

D. Effects of Confidence Level and Sample Size on Mar- gin of Error

  • Margin of Error for a confidence interval increase as the confidence level increases
  • Margin of Error for a confidence interval decreases as the sample size increases

E. Interpretation of the Confidence Level

  • The confidence level (such as 95% or 99%) gives the success rate of the confidence interval for- mula. That is if the formula is used many times for many different samples, then in the long run 95% (or 99%) of the different intervals would be successful in containing the population propor- tion p.

7.3 How Can We Construct a Confidence Inter- val to Estimate a Population Mean?

  1. The “t” distribution
    • Suppose we are random sampling from a normal population with mean μ and standard deviation σ.
  • Let n be the sample size
  • Let x be the sample mean
  • Let s be the sample standard deviation
  • The ratio (^) s/x−√μn has a probability distribution called the t distribution
  • Properties of t distributions: page 336
  1. Finding percentiles from a t distribution, Table B

3 Example. A study of the ability of individuals to walk in a straigth line. An article reported the fol- lowing data on cadence (strides per second) for a sample of n = 20 randomly selected men: 0.95 0.85 0.92 0.95 0.93 0.86 1.00 0.92 0.85 0. 0.78 0.93 0.93 1.05 0.93 1.06 1.06 0.96 0.81 0. Calculate a 99% confidence interval for the popula- tion mean cadence. From SPSS, x = 0.926, s = 0. 0809 Stem and Leaf display of data shows approximate bell shape From Table B, 0.995 percentile from t distribution with df = 20 − 1 = 19 is 2. Endpoints of confidence interval: x ± 2. 861 √s 20 After calcuation, endpoints are: 0. 926 ± 0. 052 99% confidence interval for population mean cadence is (0. 874 , 0. 978