Estimating Parameters and Determining Sample Sizes

Estimating Parameters and Determining Sample Sizes

  • This section covers the estimation of various population parameters and the determination of required sample sizes.

    • 7−1: Estimating a Population Proportion

    • 7−2: Estimating a Population Mean

    • 7−3: Estimating a Population Standard Deviation or Variance

    • 7−4: Bootstrapping: Using Technology for Estimates

Point Estimate

  • A point estimate is a single value used to estimate a population parameter.

    • The sample proportion (denoted as ( \hat{p} )) is a natural point estimate of the population proportion ( p ).

  • An unbiased estimator is a statistic whose sampling distribution has a mean equal to the corresponding population parameter.

    • The statistic ( \hat{p} ) is used as the point estimate of ( p ) because it is unbiased.

Confidence Interval (CI)

  • A confidence interval is a range of values used to estimate the true value of a population parameter.

    • Notation: CI is sometimes denoted as (A, B), where A and B are the lower and upper limits of the interval.

  • The confidence level is the probability that the confidence interval contains the population parameter.

    • Common levels include 90%, 95%, and 99%, with corresponding values of ( \alpha ) being 0.10, 0.05, and 0.01, respectively.

Relationship Between Confidence Level and ( \alpha )

    Confidence Level

    Corresponding ( \alpha )


    90%

    0.10


    95%

    0.05


    99%

    0.01

    Requirements for CI Estimating a Population Proportion

    1. The sample must be a simple random sample.

    2. Conditions for the binomial distribution must be satisfied:

      • Fixed number of trials, independent trials, two categories of outcomes, constant probabilities.

    Notation for Population Proportion CI

    • ( p = ) population proportion

    • ( \hat{p} = ) sample proportion

    • ( n = ) number of sample values

    • ( z_{\alpha/2} = ) critical value z-score that separates an area of ( \alpha/2 ) in the right tail of the standard normal distribution

    Confidence Interval Formula for Population Proportion

    • CI Estimate: [ \hat{p} - E < p < \hat{p} + E ] where [ E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} ]

      • Margin of error ( E ) can also be presented as [ (\hat{p} - E, \hat{p} + E) ]

    Critical Values

    • A critical value for a normal distribution signifies the z score separating extreme values from those that are not significant.

    • For a 95% confidence level, the critical z score corresponds to ( 0.9750 ) cumulative area from the left.

    Interpreting Confidence Intervals

    • Correct Interpretation:

      • “We are 95% confident that the true population proportion ( p ) lies within the interval.”

    • Common Misinterpretations to Avoid:

      • “There is a 95% chance that the true value is between A and B.”

      • “95% of sample proportions will fall between A and B.”

    Example: Constructing a Confidence Interval for Online Courses

    Scenario

    • A survey indicated that 53% of undergraduates take online courses.

    • Sample size ( n = 950 ).

    Steps:

    1. Margin of Error Calculation:
      [ E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} ]

      • For a 95% CI, ( z_{\alpha/2} = 1.96 ):
        [ E = 1.96 \sqrt{\frac{0.53(0.47)}{950}} ]

      • Calculate ( E \approx 0.0317381 ).

    2. Confidence Interval Construction:
      [ \hat{p} - E < p < \hat{p} + E ]

      • [ 0.53 - 0.0317381 < p < 0.53 + 0.0317381 ]

      • Result: ( 0.498 < p < 0.562 ).

      • Expressed as ( \hat{p} \pm E ) or as (0.498, 0.562).

    3. Conclusion: Cannot conclude that more than 50% of undergraduates take online courses as the CI includes values below 0.5.

    Analyzing Polls

    • When analyzing poll results:

      1. Ensure the sample is a simple random sample.

      2. Check that the confidence level is provided.

      3. Confirm the sample size information.

      4. Assess the quality of results relative to sampling method and size, rather than population size.

    Determining Sample Size

    • When planning a survey, the necessary sample size depends on:

      • Desired margin of error (E).

    Sample Size Calculation Formula

    • When ( \hat{p} ) is known:
      [ n = \frac{z_{\alpha/2}^2 \hat{p}(1 - \hat{p})}{E^2} ]

    • When ( \hat{p} ) is unknown:
      [ n = \frac{z_{\alpha/2}^2 \cdot 0.25}{E^2} ]

    Example of Required Sample Size

    • To determine opinions on online shopping:

    1. If ( \hat{p} = 0.79 ), use prior knowledge.

    2. For a 95% CI with ( E = 0.03 ):
      [ n = \frac{(1.96^2)(0.79)(0.21)}{0.03^2} = 709. \

    3. Without prior knowledge, required sample size increases to 1068.

    Summary on Bootstrapping Method

    • Bootstrapping allows estimates without strong assumptions about sample normality.

    • Key Considerations:

      1. Collect data via an appropriate method (simple random sample).

      2. Use sufficient values to provide accurate estimates.

    • The bootstrap method helps when traditional assumptions of normality are violated.

    • These notes summarize important concepts in estimating population parameters and determining sample sizes. Use them to prepare for exams and understand the underlying statistical principles.