Confidence Intervals and Sample Size in Statistics

Confidence Intervals for Proportions

Introduction

Confidence intervals are a fundamental concept in statistics, used to estimate the range within which a population parameter is likely to fall, based on sample data. This section focuses on constructing confidence intervals for population proportions.

• Population proportion (p-hat): The fraction of the sample exhibiting a particular characteristic.

• Critical z-score (Z): The value from the standard normal distribution corresponding to the desired confidence level (e.g., 1.96 for 95%).

• Margin of error (E): The maximum expected difference between the true population parameter and a sample estimate.

Key Formulas

• Sample proportion: $\hat{p} = \frac{x}{n}$, where x is the number of successes and n is the sample size.

• Complement of sample proportion: $\hat{q} = 1 - \hat{p}$

• Margin of error for proportion: $E = Z \sqrt{\frac{\hat{p}\hat{q}}{n}}$

• Confidence interval: $\hat{p} - E < p < \hat{p} + E$

Example

If a survey of 100 people finds that 55 eat breakfast every morning:

• $\hat{p} = \frac{55}{100} = 0.55$

• $\hat{q} = 1 - 0.55 = 0.45$

• For 95% confidence, $Z = 1.96$

• $E = 1.96 \sqrt{\frac{0.55 \times 0.45}{100}} = 1.96 \sqrt{0.002475} = 1.96 \times 0.0497 = 0.098$

• Confidence interval: $0.55 - 0.098 < p < 0.55 + 0.098$ or $0.452 < p < 0.648$

Interpretation: We are 95% confident that the true proportion of people who eat breakfast every morning is between 45.2% and 64.8%.

Practice Question

• Given a sample of 100 people, 60 drink coffee every morning. Find the 95% confidence interval for this proportion.

• Steps:

  1. Calculate $\hat{p}$ and $\hat{q}$

  2. Find the critical z-score

  3. Compute the margin of error

  4. State the confidence interval

Determining Sample Size for a Desired Margin of Error

Introduction

To ensure that a sample estimate is within a specified margin of error at a given confidence level, it is necessary to calculate the required sample size before conducting a survey or experiment.

• Desired margin of error (E): The maximum acceptable difference between the sample estimate and the true population value.

• Estimated proportion (p): If unknown, use 0.5 for maximum variability.

Key Formula

• Sample size for proportion: $n = \frac{Z^2 \hat{p} \hat{q}}{E^2}$

Example

To be within 4 percentage points (E = 0.04) at 95% confidence, assuming $\hat{p} = 0.25$:

• $n = \frac{1.96^2 \times 0.25 \times 0.75}{0.04^2} = \frac{3.8416 \times 0.1875}{0.0016} = \frac{0.7203}{0.0016} = 450.2$

• Round up: Sample size needed is 451

Practice Question

• Find the sample size needed to be within 5 percentage points (E = 0.05) at 95% confidence, assuming no prior knowledge of the proportion.

Confidence Intervals for Means (Using t-distribution)

Introduction

When estimating the mean of a population from sample data, especially with small samples or unknown population standard deviation, the t-distribution is used to construct confidence intervals.

• Sample mean (x̄): The average value from the sample.

• Sample standard deviation (s): Measures the spread of sample data.

• Degrees of freedom (df): $df = n - 1$, where n is the sample size.

• Critical t-value (tα/2): Value from the t-distribution table for the desired confidence level and degrees of freedom.

• Margin of error (E): $E = t_{\alpha/2} \frac{s}{\sqrt{n}}$

Key Formula

• Confidence interval for mean: $\bar{x} - E < \mu < \bar{x} + E$

Example

Sample size = 40, sample mean = 30, sample standard deviation = 4, 95% confidence:

• Degrees of freedom: $df = 40 - 1 = 39$

• Critical t-value (from table): $t_{0.025,39} = 2.021$

• Margin of error: $E = 2.021 \times \frac{4}{\sqrt{40}} = 2.021 \times 0.632455 = 1.278$

• Confidence interval: $30 - 1.278 < \mu < 30 + 1.278$ or $28.722 < \mu < 31.278$

Practice Question

• Construct a confidence interval for a mean of 20, sample size 49, sample standard deviation 3.

Summary Table: Confidence Interval Formulas

Parameter	Formula	Distribution
Proportion	$\hat{p} \pm Z \sqrt{\frac{\hat{p}\hat{q}}{n}}$	Normal (z)
Mean (unknown population SD)	$\bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}}$	t-distribution
Sample size for proportion	$n = \frac{Z^2 \hat{p} \hat{q}}{E^2}$	Normal (z)

Additional info:

• When the population proportion is unknown, use $\hat{p} = 0.5$ for maximum sample size.

• Always round up sample size calculations to ensure the margin of error is not exceeded.

• For small samples (typically n < 30), use the t-distribution for means.