Chapter 9: Confidence Intervals for Population Parameters

Confidence Intervals for Population Parameters

Introduction to Confidence Intervals

This chapter focuses on constructing confidence intervals to estimate various population parameters, including the proportion, mean, and standard deviation (or variance). Confidence intervals provide a range of plausible values for an unknown parameter, based on sample data. Understanding how to construct and interpret these intervals is fundamental in inferential statistics.

• Parameter: A numerical summary describing a population (e.g., population mean μ, population proportion p).

• Statistic: A numerical summary describing a sample (e.g., sample mean \( \bar{x} \), sample proportion \( \hat{p} \)).

• Point Estimate: The value of a statistic used to estimate a population parameter.

• Confidence Interval: An interval of numbers, centered at the point estimate, that is likely to contain the population parameter.

• Level of Confidence: The probability (expressed as a percentage) that the confidence interval contains the true parameter value in repeated sampling. Common levels: 90%, 95%, 99%.

• Significance Level (\( \alpha \)): The probability of error we are willing to accept (commonly 0.05, 0.01, or 0.10). The level of confidence is \( (1 - \alpha) \times 100\% \).

• Margin of Error: The amount added and subtracted from the point estimate to create the confidence interval.

General Form of a Confidence Interval:

• \( \text{Point Estimate} \pm \text{Margin of Error} \)

Example: If the average grade in a class is 81.9, a possible interval estimate could be 81.9 ± 15.1, or (66.8, 97). The margin of error (15.1) is calculated using methods described in this chapter.

Estimating a Population Proportion

Section 9.1: Confidence Interval for a Population Proportion

When estimating a population proportion, we use the sample proportion (\( \hat{p} \)) as the point estimate. The confidence interval provides a range of plausible values for the true population proportion (p).

• Sample Proportion (\( \hat{p} \)): The proportion of successes in the sample.

• Normal Approximation: For large samples, the sampling distribution of \( \hat{p} \) is approximately normal (by the Central Limit Theorem), allowing us to use the normal (z) distribution to construct the interval.

• Confidence Interval Formula:

$\hat{p} \pm z_{\alpha/2} \sqrt{ \frac{ \hat{p}(1 - \hat{p}) }{ n } }$

• Where \( z_{\alpha/2} \) is the critical value from the standard normal distribution for the desired confidence level.

• \( n \) is the sample size.

Sample Size Formula: To determine the required sample size for a desired margin of error (E):

$n = \frac{ z_{\alpha/2}^2 \cdot \hat{p}(1 - \hat{p}) }{ E^2 }$

Example Application: Suppose you want to estimate the proportion of students who pass an exam with 95% confidence and a margin of error of 0.05. Use the formula above to determine the necessary sample size.

Estimating a Population Mean

Section 9.2: Confidence Interval for a Population Mean

When estimating a population mean, the method depends on whether the population standard deviation (σ) is known. If σ is unknown (the usual case), we use the sample standard deviation (s) and the Student's t-distribution.

• Student's t-Distribution: Used when the population standard deviation is unknown and the sample is drawn from a normally distributed population.

• Degrees of Freedom (df): For a sample of size n, df = n - 1.

• Confidence Interval Formula:

$\bar{x} \pm t_{\alpha/2,\,df} \left( \frac{s}{\sqrt{n}} \right)$

• Where \( \bar{x} \) is the sample mean, s is the sample standard deviation, n is the sample size, and \( t_{\alpha/2,\,df} \) is the critical value from the t-distribution.

Example: Given n = 20, \( \bar{x} = 50 \), s = 8, and a 98% confidence level:

• \( \alpha = 0.02 \), \( \alpha/2 = 0.01 \)

• \( t_{0.01, 19} = 2.539 \) (from t-table)

• Margin of error: $2.539 \times \frac{8}{\sqrt{20}} \approx 4.542$

• Confidence interval: (50 - 4.542, 50 + 4.542) = (45.5, 54.5)

Sample Size Formula: To achieve a desired margin of error (E):

$n = \left( \frac{ z_{\alpha/2} \cdot \sigma }{ E } \right)^2$

Additional info: If σ is unknown, use a preliminary estimate or the sample standard deviation from a pilot study.

Estimating a Population Standard Deviation or Variance

Section 9.3: Confidence Interval for a Population Standard Deviation or Variance

To estimate the population standard deviation (σ) or variance (σ²), we use the Chi-Square (χ²) distribution. This method is appropriate when the population is normally distributed.

• Chi-Square Distribution: A family of distributions used to model the distribution of the sum of squared standard normal variables. The Greek letter χ (chi) is used, not the letter x.

• Degrees of Freedom (df): For a sample of size n, df = n - 1.

• Confidence Interval for Variance (σ²):

$\left( \frac{ (n-1)s^2 }{ \chi^2_{\alpha/2,\,df} }, \frac{ (n-1)s^2 }{ \chi^2_{1-\alpha/2,\,df} } \right)$

• Where s² is the sample variance, and \( \chi^2_{\alpha/2,\,df} \) and \( \chi^2_{1-\alpha/2,\,df} \) are critical values from the chi-square distribution.

• To find these values, use a chi-square table (Table VIII).

Confidence Interval for Standard Deviation (σ): Take the square root of the interval endpoints for variance.

Example Application: Suppose n = 15, s = 4.2. To construct a 95% confidence interval for σ², find the appropriate chi-square critical values for df = 14 and apply the formula above.

Summary and Additional Notes

Section 9.4: Summary of Methods

This section provides a concise summary of the confidence interval methods for different parameters. Focus on understanding the formulas and when to use each distribution (z, t, or chi-square).

• Use the z-distribution for proportions and means when σ is known and n is large.

• Use the t-distribution for means when σ is unknown and the sample is from a normal population.

• Use the chi-square distribution for variances and standard deviations.

Rounding: Proper rounding is essential for accurate answers, especially in online homework systems. Follow the specific rounding rules provided in your course materials.

Technology: Software such as Minitab and StatCrunch can be used to construct confidence intervals, but manual calculation is also important for understanding the process.

Section 9.5: Additional Reading

This section is primarily a reading assignment and is not emphasized in exams or homework.