Confidence Interval Estimation: Concepts, Calculations, and Applications

Chapter 7: Confidence Interval Estimation

Introduction to Confidence Interval Estimation

Confidence interval estimation is a fundamental concept in statistics used to estimate population parameters based on sample data. It provides a range of plausible values for the parameter, along with a specified level of confidence.

• Parameter: The true value of interest in the population, such as the average in-state tuition fee per semester at a university (μ).

• Types of Estimates: There are two main types of estimates for a parameter:

Types of Estimates	Example Value
Point Estimate	$2,300
Confidence Interval Estimate	$2,000 to $3,000 with 95% confidence

The subject of this chapter is the confidence interval estimate.

Preliminaries: Revisiting the Empirical Rule

The empirical rule helps us understand the distribution of data in a normal distribution. For confidence intervals, we often use the standard normal distribution (z-distribution).

• Empirical Rule Tweaked:

• $P(-2 \leq z \leq 2) = normsdist(2) - normsdist(-2) \approx 0.9545$

• $P(-1.96 \leq z \leq 1.96) = normsdist(1.96) - normsdist(-1.96) \approx 0.9500$

• The value 1.96 is commonly used for 95% confidence intervals.

Population Mean and Standard Deviation

In many cases, the population mean may change, but the standard deviation remains relatively constant. This is illustrated by comparing different populations:

Population	Mean	SD
Population 1	7.5	1.7
Population 2	8.5	1.7
Population 3	12.5	1.7
Population 4	22.5	1.7

Key Point: The mean can vary, but the standard deviation often remains stable across similar populations.

Quantitative vs. Qualitative Variables

• Quantitative Variables: Measurements are mean and standard deviation.

• Qualitative Variables: Measurement is proportion.

Confidence Interval for Population Mean

Case 1: Population Standard Deviation Known

When the population standard deviation (σ) is known, the confidence interval for the mean is calculated using the z-distribution.

• Draw a simple random sample of size n from the population.

• Calculate the sample mean, X̄.

• Point Estimate: $\mu \approx \bar{X}$

• 95% Confidence Interval: $\bar{X} \pm \text{MoE}$, where Margin of Error (MoE) is:

$\text{MoE} = Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

• $Z_{\alpha/2}$ is the critical value from the standard normal distribution for the desired confidence level.

Confidence Level	$\alpha$	$\alpha/2$	$1-\alpha/2$	$Z_{\alpha/2}$
90%	0.10	0.050	0.950	1.65
95%	0.05	0.025	0.975	1.96
99%	0.01	0.005	0.995	2.58

• Example: A sample of 40 students has an average ACT score of 28, with a known population SD of 5. For a 99% confidence interval:

$\text{MoE} = 2.58 \times \frac{5}{\sqrt{40}} \approx 2.0$

99% CI: $28 \pm 2.0 = [26, 30]$

Interpretation: "We are 99% confident that the actual average ACT score for students on that campus is between 26 and 30 points."

Case 2: Population Standard Deviation Unknown

When the population standard deviation is unknown, we estimate it using the sample standard deviation (S) and use the t-distribution.

• Estimate SD by sample SD, S.

• Use the t-distribution with n-1 degrees of freedom (df).

• Formula: $\bar{X} \pm t_{\alpha/2, n-1} \frac{S}{\sqrt{n}}$

• $t_{\alpha/2, n-1}$ is the critical value from the t-distribution for the desired confidence level and degrees of freedom.

• Properties of the t-distribution:

• Bell-shaped and centered at 0, like the normal distribution.

• Depends on sample size (df); infinite number of t-distributions.

• As df increases, t-distribution approaches the z-distribution.

• t-distribution has more area in the tails compared to the z-distribution.

• Example: Sample of 6 students' GPA: 3.20, 3.75, 3.10, 2.95, 3.40. Sample mean = 3.20, sample SD = 0.34, df = 5.

• 95% CI: $3.20 \pm 2.571 \times \frac{0.34}{\sqrt{6}} \approx 3.20 \pm 0.36 = [2.84, 3.56]$

• Interpretation: "We are 95% confident that the actual average GPA of the students is between 2.84 and 3.56."

Confidence Interval for Population Proportion

Estimating a Proportion

For qualitative variables, the parameter of interest is the proportion (P).

• Draw a simple random sample of size n.

• Sample estimate of P: $\hat{p} = \frac{x}{n}$, where x is the number of successes.

• Confidence Interval: $\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Valid if $n\hat{p} > 5$ and $n(1-\hat{p}) > 5$.

• Example: Newspaper survey: 656 out of 1600 want more local news. $\hat{p} = 656/1600 = 0.41$

• 95% CI: $0.41 \pm 1.96 \times \sqrt{\frac{0.41 \times 0.59}{1600}} \approx 0.41 \pm 0.024 = [0.386, 0.434]$ or [38.6%, 43.4%]

• Interpretation: "We are 95% confident that the percentage of readers who want more local news is between 39% and 43%."

Sample Size Determination

For Estimating the Mean

To achieve a desired margin of error (ME) for a confidence interval, the required sample size can be calculated as:

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

• The smaller the margin of error, the larger the sample size required.

• The larger the confidence level or population SD, the larger the sample size required.

• Example: To estimate average internet usage with 90% confidence and ME of 2 minutes, $\sigma = 15$:

• $n = \left( \frac{1.65 \times 15}{2} \right)^2 \approx 153$

For Estimating a Proportion

To achieve a desired margin of error (ME) for a proportion:

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

• Example: For 95% CI, $\hat{p} = 0.35$, ME = 0.03:

• $n = \frac{(1.96)^2 \times 0.35 \times 0.65}{0.03^2} \approx 977$

Effect of Margin of Error: Smaller ME requires a larger sample size.

Summary Table: Key Formulas

Parameter	CI Formula	Distribution
Mean (σ known)	$\bar{X} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$	z-distribution
Mean (σ unknown)	$\bar{X} \pm t_{\alpha/2, n-1} \frac{S}{\sqrt{n}}$	t-distribution
Proportion	$\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	z-distribution

Additional info: All formulas assume random sampling and normality conditions (or large sample size for proportions).