Confidence Intervals and Sample Size Notes

Overview of Confidence Intervals and Sample Size

  • Chapter Objective: Construct confidence intervals and determine minimum sample sizes for population means and proportions.

Key Concepts

  • Confidence Interval: An interval estimate of a parameter that likely contains the true value of that parameter with a specified confidence level.

  • Confidence Level: The probability that the interval estimate will contain the population parameter.

    • For example, a 90% confidence level means there is a 90% chance the interval contains the true population mean.

Estimation

  • Estimation: The process of using sample data to estimate true population parameters.

  • Types of Estimates:

    1. Point Estimate: A single value estimate of a parameter.

    2. Interval Estimate: A range of values used to estimate a parameter.

Properties of a Good Estimator

  1. Unbiased: The expected value equals the parameter.

    • Example: The sample mean is an unbiased estimator of the population mean.

  2. Consistent: As sample size increases, the estimator approaches the true parameter value.

  3. Relatively Efficient: The estimator with the smallest variance among all unbiased estimators is preferred.

Constructing Confidence Intervals for Population Mean

When Population Standard Deviation (σ) is Known

  • Formula for Confidence Interval:
    ar{X} o z imes rac{ rac{ au}{ ext{n}}}{ au}

    • Where:

    • $ar{X}$ = sample mean

    • $z$ = z-score corresponding to the confidence level

    • $ au$ = population standard deviation

    • $n$ = sample size

  • Confidence Levels and z-scores:

    • 99%: $z = 2.58$

    • 95%: $z = 1.96$

    • 90%: $z = 1.65$

  • Example:

    • Sample mean spending per visit at bookstore is $23.45.

    • 99% confidence interval results in an interval between $22.42 and $24.48.

When Population Standard Deviation (σ) is Unknown

  • Use t-distribution instead of z-distribution.

  • Degrees of Freedom (df): $df = n - 1$.

  • Confidence Interval Formula: ar{X} o t_{a/2} imes rac{s}{ ext{n}}

    • Where $s$ is the sample standard deviation.

Determining Minimum Sample Size

  • Formula for Minimum Sample Size (n):
    n = igg( rac{z imes au}{E}igg)^{2}

    • Where:

    • $E$ = margin of error

    • $z$ = z-value for confidence level

    • $ au$ = population standard deviation

  • Example Calculation for Pizza Shop:

    • Desired margin of error: $0.15;

    • Past study σ: $0.26;

    • Calculated sample size needed: 12.

Confidence Intervals for Proportion

  • Population Proportion (p): The ratio of units with a characteristic.
    p = rac{ ext{X}}{N}

    • Where:

    • $X$ = number of favorable responses;

    • $N$ = total sample size.

  • Confidence Interval for Population Proportion:
    ext{CI} = p o z{a/2} imes rac{ rac{ au{p}}{ ext{n}}}{ au_{p}}

    • This requires estimation of the proportion and for large sample sizes, will approach normality.

  • Example Calculation for Fiji Study:

    • 100 people surveyed, 27 obese yields a 95% confidence interval for the proportion to be between 18.3% - 35.7%.

Summary

  • This chapter highlighted the construction of confidence intervals for means and proportions, estimation, properties of estimators, and determining appropriate sample sizes.

Exercises

  1. Define confidence level and confidence interval.

  2. Calculate 95% CI for given data points.

  3. Sample size estimation based on variance and confidence.

  4. Analyze heart beats data and determine point estimate and confidence interval.

  5. Estimate true mean in survey context and apply methodologies from the chapter.

Confidence intervals (CIs) estimate population parameters, with a specified confidence level indicating the probability that the interval contains the true parameter. Estimation uses sample data for true parameter estimates, involving point (single value) and interval (range) estimates. A good estimator is unbiased, consistent, and relatively efficient. To construct CIs for the population mean: when standard deviation (σ) is known, use ar{X} ext{ ± } z imes rac{ au}{ ext{n}}; when σ is unknown, use the t-distribution: ar{X} ext{ ± } t{a/2} imes rac{s}{ ext{n}}. The minimum sample size formula is n = igg( rac{z imes au}{E}igg)^{2}. For proportions, compute p = rac{X}{N} and CI as ext{CI} = p ext{ ± } z{a/2} imes rac{ au p}{ ext{n}}. This chapter covers confidence intervals' construction, estimation processes, properties of estimators, and determining sample sizes.