Estimates and Sample Sizes Notes

Chapter 6: Estimates and Sample Sizes

6-1 Review and Preview

  • Inferential statistics involves two main activities:
  1. Using sample data to estimate population parameters
  2. Testing hypotheses about population parameters
  • This chapter covers methods for estimating important population parameters:
  • Proportions
  • Means
  • Variances
  • Methods for determining necessary sample sizes for these estimations are also introduced.

6-2 Estimating a Population Proportion

  • Key Concept: Use sample proportion to estimate the value of population proportion.
  • The sample proportion is considered the best point estimate.
  • Confidence intervals can be constructed to estimate the true value of the population proportion.
  • Knowing how to calculate required sample sizes for estimating population proportions is essential.
  • Definitions:
  • Point Estimate: A single value used to approximate a population parameter (e.g., sample proportion ( \hat{p} ) ).
  • Confidence Interval (CI): A range of values estimating the true value of a population parameter.
  • Confidence Level: Probability that a CI contains the population parameter (e.g., 90%, 95%, 99%).

Example

  • Finding Best Estimate: In a Pew Research Center poll, 70% of 1,501 adults believe in global warming. The best point estimate of ( p ) is 0.70.

Interpreting Confidence Intervals

  • CI example interpretation: “We are 95% confident that the interval from 0.677 to 0.723 contains the true population proportion.”
  • Critical Value ( z_{\alpha/2} ): Value separating likely and unlikely sample statistics. Available using z-tables for specified confidence levels (e.g., 1.96 for 95%).
  • Margin of Error (E): The difference between the sample proportion and the true population proportion, calculated using the formula:
    [ E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} ]
    where n is the sample size.

Procedure for Constructing CI for Proportion

  1. Verify that the required assumptions are satisfied.
  2. Find the critical value for the desired confidence level.
  3. Evaluate the margin of error ( E ).
  4. Compute the confidence interval: ( \hat{p} - E < p < \hat{p} + E ).

6-3 Estimating a Population Mean (σ Known)

  • Methods for estimating a population mean when the population standard deviation ( \sigma ) is known.
  • Key Concepts:
  1. Sample mean ( \bar{x} ) is the best point estimate of population mean ( \mu ).
  2. Confidence intervals can be constructed for the population mean using sample data.
  3. Necessary sample size determination is crucial for accurate estimation.
  • CI for Mean with Known ( \sigma ): ( \mu ) is estimated with
    [ \bar{x} - E < \mu < \bar{x} + E ]
    where ( E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}} ).

6-4 Estimating a Population Mean (σ Not Known)

  • Use the Student t Distribution when the population standard deviation is unknown.
  • Key Concept: The t distribution accounts for additional variability expected with small samples.
  • Margin of Error with t Distribution:
    [ E = t{\alpha/2} \frac{s}{\sqrt{n}} ] where ( t{\alpha/2} ) corresponds to the degrees of freedom ( n - 1 ).
  • Construct CI for mean with unknown ( \sigma ):
    [ \bar{x} - E < \mu < \bar{x} + E ]

6-5 Estimating a Population Variance

  • Chi-Square Distribution is introduced to construct confidence intervals for population standard deviation and variance.
  • To estimate the population variance, the chi-square statistic is used:
    [ \chi^2 = \frac{(n - 1)s^2}{\sigma^2} ]
    where ( s^2 ) is the sample variance.

Confidence Interval for Variance

  1. Verify that a simple random sample has been taken, and the population is normally distributed.
  2. Use the critical values from the chi-square table for necessary bounds.
  3. Evaluate the confidence limits and take square root for standard deviation if needed.