Confidence Intervals for Proportions and Means

Confidence Intervals

Introduction

Confidence intervals are a fundamental concept in statistics, used to estimate population parameters (such as proportions or means) based on sample data. They provide a range of plausible values for the parameter, along with a specified level of confidence (commonly 95%).

Confidence Intervals for Proportions

Key Concepts and Definitions

• Proportion (p-hat, \(\hat{p}\)): The sample proportion, calculated as the number of successes divided by the sample size.

• Complement of Proportion (q-hat, \(\hat{q}\)): The proportion of failures, calculated as \(1 - \hat{p}\).

• Sample Size (n): The number of observations in the sample.

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

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

Formula for Confidence Interval for a Proportion

The confidence interval for a population proportion is calculated as:

$\hat{p} \pm Z \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$

• \(\hat{p}\) = sample proportion

• \(\hat{q}\) = 1 - \(\hat{p}\)

• n = sample size

• Z = critical value for the desired confidence level

Example: Calculating a 95% Confidence Interval for a Proportion

• Suppose a simple random sample of 100 people is surveyed, and 55 of them eat breakfast every morning.

• \(\hat{p} = 55/100 = 0.55\)

• \(\hat{q} = 1 - 0.55 = 0.45\)

• n = 100

• Z = 1.96 (for 95% confidence)

• Margin of error: $E = 1.96 \times \sqrt{\dfrac{0.55 \times 0.45}{100}}$ $E = 1.96 \times 0.049497$ $E = 0.0968$ (rounded to 3 significant places)

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

Interpretation: With 95% confidence, the actual percentage of the population who eat breakfast every morning lies between 45.2% and 64.8%.

Sample Size Determination for Proportions

To determine the required sample size for a desired margin of error (E) and confidence level, use:

$n = \dfrac{Z^2 \hat{p} \hat{q}}{E^2}$

• If no prior knowledge of \(\hat{p}\), use 0.25 (maximum variability: \(\hat{p} = 0.5, \hat{q} = 0.5\)).

Example: Sample Size Calculation

• Desired margin of error: 4 percentage points (0.04)

• Confidence level: 95% (Z = 1.96)

• Assume \(\hat{p} = 0.5, \hat{q} = 0.5\)

• $n = \dfrac{1.96^2 \times 0.25}{0.04^2}$ $n = \dfrac{3.8416 \times 0.25}{0.0016}$ $n = \dfrac{0.9604}{0.0016}$ $n = 600.25$ Round up to 601.

Interpretation: To be 95% confident that the sample percentage is within 4 percentage points of the true population percentage, a sample of at least 601 is needed.

Confidence Intervals for Means

Key Concepts and Definitions

• Sample Mean (\(\bar{x}\)): The average value in the sample.

• Sample Standard Deviation (s): A measure of the spread of sample values.

• Sample Size (n): Number of observations in the sample.

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

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

• Margin of Error (E): The maximum expected difference between the sample mean and the true population mean.

Formula for Confidence Interval for a Mean

The confidence interval for a population mean is calculated as:

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

• \(\bar{x}\) = sample mean

• s = sample standard deviation

• n = sample size

• $t_{\alpha/2}$ = critical t-value for the desired confidence level and degrees of freedom

Example: Calculating a 95% Confidence Interval for a Mean

• Sample mean (\(\bar{x}\)) = 30

• Sample standard deviation (s) = 4

• Sample size (n) = 40

• Degrees of freedom = 39

• Critical t-value (for 95% confidence, df = 40): $t_{0.025,40} = 2.021$

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

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

Interpretation: With 95% confidence, the population mean lies between 28.722 and 31.278.

Table: Summary of Confidence Interval Formulas

Parameter	Formula	Critical Value	When to Use
Proportion	$\hat{p} \pm Z \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$	Z (from normal distribution)	Large n, estimating a proportion
Mean	$\bar{x} \pm t_{\alpha/2} \dfrac{s}{\sqrt{n}}$	t (from t-distribution)	Unknown population standard deviation, small/medium n

Practice Questions (from the file)

1. You survey a simple random sample of 100 people and 60 of them drink coffee every morning. Create a confidence interval at a 95% confidence level for this proportion.

   • What is the critical z-score in this example?

   • What is p-hat?

   • What is q-hat?

   • What is the margin of error - E?

   • What is the confidence interval?

   • What statement can you make based on the above?

2. If you want to be within 5 percentage points for your margin of error (E) at a 95% confidence level, and you have no prior knowledge of a sample proportion, how large should your simple random sample of adults be?

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

Additional info: The notes also mention using the largest possible value for \(\hat{p}\) (0.5) when no prior knowledge is available, as this maximizes the required sample size for a given margin of error.