Chapter 9: Confidence Intervals for Population Parameters

Confidence Intervals for Population Parameters

Introduction to Confidence Intervals

Confidence intervals are a fundamental concept in inferential statistics, allowing us to estimate population parameters (such as proportion, mean, or standard deviation) based on sample data. This chapter focuses on constructing and interpreting confidence intervals for different types of parameters, emphasizing both manual calculation and the use of statistical software.

• 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 values, derived from a sample, that is likely to contain the population parameter with a specified level of confidence.

• 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 (e.g., 0.05 for 95% confidence).

• General Form: $\text{Point Estimate} \pm \text{Margin of Error}$

Example: If the average grade in a class is 81.9, a possible confidence interval might be (66.8, 97), or 81.9 ± 15.1. The margin of error (15.1) is calculated based on the desired confidence level and sample variability.

Estimating a Population Proportion

Constructing Confidence Intervals for Proportions

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 observed in the sample.

• Confidence Interval Formula:

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

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

• Assumptions: The sample size is large enough for the normal approximation to be valid (np ≥ 5 and n(1-p) ≥ 5).

• Level of Confidence: Calculated as (1 – α) × 100% (e.g., for α = 0.05, confidence level is 95%).

Example: If \( \hat{p} = 0.6 \), n = 100, and for 95% confidence, \( z_{0.025} = 1.96 \):

$0.6 \pm 1.96 \sqrt{ \dfrac{0.6 \times 0.4}{100} } = 0.6 \pm 0.096$ So, the confidence interval is (0.504, 0.696).

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

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

Additional info: If p is unknown, use p = 0.5 for a conservative estimate.

Estimating a Population Mean

Confidence Intervals Using the t-Distribution

When estimating a population mean and the population standard deviation is unknown, the t-distribution is used. This accounts for additional uncertainty due to estimating the standard deviation from the sample.

• t-Distribution: A probability distribution used when estimating the mean of a normally distributed population with unknown standard deviation, especially for small samples (n < 30).

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

• Confidence Interval Formula:

$\bar{x} \pm t_{\alpha/2,\,df} \dfrac{s}{\sqrt{n}}$

• 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, 98% confidence:

• α = 0.02, α/2 = 0.01, \( t_{0.01,19} = 2.539 \)

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

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

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

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

Additional info: If σ is unknown, use an estimate from prior studies or a pilot sample.

Estimating a Population Standard Deviation or Variance

Confidence Intervals Using the Chi-Square Distribution

To estimate the population standard deviation (σ) or variance (σ²), the chi-square (χ²) distribution is used. This method is appropriate when the population is normally distributed.

• Chi-Square Distribution: A distribution used to model the sum of squared standard normal variables. It is not symmetric and depends on degrees of freedom (df = n – 1).

• Key Properties:

  • Values are always positive.

  • Shape depends on degrees of freedom.

  • Used for inference about variance and standard deviation.

• Confidence Interval for Variance (σ²):

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

• Where s² is the sample variance, n is the sample size, 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).

Example: Suppose n = 15, s² = 25, 95% confidence. Find \( \chi^2_{0.025,14} \) and \( \chi^2_{0.975,14} \) from the table, then compute the interval using the formula above.

Summary of Chapter 9 Methods

Overview Table: Confidence Interval Methods

The following table summarizes the main methods for constructing confidence intervals for different parameters:

Parameter	Point Estimate	Distribution	Confidence Interval Formula
Proportion (p)	\( \hat{p} \)	Normal (z)	$\hat{p} \pm z_{\alpha/2} \sqrt{ \dfrac{ \hat{p}(1-\hat{p}) }{ n } }$
Mean (μ)	\( \bar{x} \)	t (if σ unknown)	$\bar{x} \pm t_{\alpha/2,\,df} \dfrac{s}{\sqrt{n}}$
Variance (σ²)	\( s^2 \)	Chi-Square	$\left( \dfrac{ (n-1)s^2 }{ \chi^2_{\alpha/2,\,df} }, \dfrac{ (n-1)s^2 }{ \chi^2_{1-\alpha/2,\,df} } \right)$

Additional info: Section 9.4 provides a concise summary of these methods, while Section 9.5 is a reading assignment and not emphasized for exams.