7.1

Estimating Parameters and Determining Sample Sizes

Overview

This chapter introduces statistical methods for estimating population parameters using sample data, with a focus on population proportions. It covers point estimation, confidence intervals, and determining appropriate sample sizes for reliable inference.

Estimating a Population Proportion

Key Concepts

• Point Estimate: The sample proportion ($\hat{p}$) is the best point estimate of the population proportion $p$.

• Confidence Interval: An interval estimate constructed from sample data to estimate the true value of a population proportion.

• Sample Size: The number of observations required to estimate a population proportion with a specified level of confidence and margin of error.

Point Estimate

A point estimate is a single value used to estimate a population parameter. For proportions, the sample proportion ($\hat{p}$) is used because it is unbiased and consistent.

• Unbiased: The expected value of $\hat{p}$ equals the true population proportion $p$.

• Consistent: As sample size increases, $\hat{p}$ tends to get closer to $p$.

Example: In a survey of 950 undergraduate students, 53% reported taking online courses. The best point estimate of the proportion of all undergraduates who take online courses is $\hat{p} = 0.53$.

Disadvantage of Point Estimate

• Point estimates are almost never exactly equal to the true population value.

• It is necessary to describe the likely error or uncertainty associated with the estimate.

Confidence Interval

A confidence interval (CI) is a range of values, derived from sample statistics, that is likely to contain the true value of a population parameter.

• General form: $\text{estimate} - \text{error} < \text{population} < \text{estimate} + \text{error}$

• Expressed as $\hat{p} \pm E$ or $(\hat{p} - E, \hat{p} + E)$

Confidence Level

The confidence level (e.g., 95%) is the probability that the confidence interval contains the true population parameter, assuming the estimation process is repeated many times.

• Denoted as $1 - \alpha$ (e.g., $\alpha = 0.05$ for 95% confidence)

• Also called the degree of confidence or confidence coefficient

• Does not mean that a specific interval has a 95% chance of containing $p$; $p$ is fixed, and the interval either contains it or not.

Relationship Between Confidence Level and Alpha

Confidence Level	Alpha ($\alpha$)
90%	0.10
95%	0.05
99%	0.01

Interpreting a Confidence Interval

• Correct: "We are 95% confident that the interval from 0.405 to 0.455 contains the true value of the population proportion $p$."

• Incorrect: "There is a 95% chance that $p$ is between 0.405 and 0.455." (Incorrect because $p$ is fixed.)

• Incorrect: "95% of sample proportions will fall between 0.405 and 0.455." (Incorrect because the interval is about the population parameter, not the sample.)

The Process Success Rate

With a 95% confidence level, the process of constructing intervals will capture the true population parameter in 95% of repeated samples.

Confidence Interval for Estimating a Population Proportion $p$

Notation

• $p$ = population proportion

• $\hat{p}$ = sample proportion

• $n$ = sample size

• $E$ = margin of error

• $z_{\alpha/2}$ = critical value (z-score for the desired confidence level)

Requirements

1. The sample is a simple random sample.

2. The binomial distribution conditions are satisfied: fixed number of independent trials, two outcomes, constant probability.

3. At least 5 successes and 5 failures in the sample.

Confidence Interval Formula

$\hat{p} - E < p < \hat{p} + E$ where $E = z_{\alpha/2} \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$ and $\hat{q} = 1 - \hat{p}$

Margin of Error

The margin of error ($E$) is the maximum likely difference between the sample statistic and the population parameter, derived from the sampling distribution.

Formula:

$E = z_{\alpha/2} \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$

• Critical value: $z_{\alpha/2}$, the z-score corresponding to the desired confidence level.

• Estimated standard deviation: $\sqrt{\dfrac{\hat{p}\hat{q}}{n}}$

Critical Values

Critical values are z-scores that separate the central area from the tails in the standard normal distribution, corresponding to the desired confidence level.

Confidence Level	Alpha ($\alpha$)	Critical Value ($z_{\alpha/2}$)
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.575

Procedure for Constructing a Confidence Interval for $p$

1. Verify that the requirements are satisfied.

2. Find the critical value $z_{\alpha/2}$ for the desired confidence level.

3. Calculate the margin of error $E = z_{\alpha/2} \sqrt{\dfrac{\hat{p}\hat{q}}{n}}$.

4. Compute the confidence interval limits: $\hat{p} - E$ and $\hat{p} + E$.

5. Round the interval limits to three significant digits.

Example: Constructing a Confidence Interval (Online Courses)

• Given: $n = 950$, $\hat{p} = 0.53$

• Find: Margin of error $E$ for 95% confidence

• Solution:

$E = 1.96 \sqrt{\dfrac{(0.53)(0.47)}{950}} = 0.0317381$

Confidence interval: $0.53 - 0.0317381 < p < 0.53 + 0.0317381$ $0.498 < p < 0.562$

• Expressed as $0.53 \pm 0.0317$ or $(0.498, 0.562)$

• Interpretation: We are 95% confident that the true proportion of undergraduates taking online courses is between 49.8% and 56.2%.

Finding the Point Estimate and Margin of Error from a Confidence Interval

• Point estimate: $\dfrac{\text{Upper limit} + \text{Lower limit}}{2}$

• Margin of error: $\dfrac{\text{Upper limit} - \text{Lower limit}}{2}$

Determining Sample Size

Overview

To estimate a population proportion with a specified margin of error and confidence level, it is necessary to determine the required sample size ($n$).

Sample Size Formula for Estimating a Population Proportion

• When an estimate of $\hat{p}$ is known:

$n = \left[ \dfrac{z_{\alpha/2}^2 \hat{p} \hat{q}}{E^2} \right]$

• When no estimate of $\hat{p}$ is known: Use $\hat{p} = 0.5$ (maximizes $\hat{p}\hat{q}$ and gives the largest required $n$)

$n = \left[ \dfrac{z_{\alpha/2}^2 (0.25)}{E^2} \right]$

• Round up to the next whole number to ensure the sample size is sufficient.

Example: Determining Sample Size for Online Purchases

• Given: Prior survey: $\hat{p} = 0.79$, $E = 0.03$, 95% confidence ($z_{\alpha/2} = 1.96$)

• Calculation:

$n = \dfrac{(1.96)^2 (0.79)(0.21)}{(0.03)^2} \approx 709$

• No prior estimate: Use $\hat{p} = 0.5$

$n = \dfrac{(1.96)^2 (0.25)}{(0.03)^2} \approx 1068$

• Interpretation: Without prior knowledge, a larger sample is required to achieve the same margin of error and confidence level.

Summary Table: Confidence Levels, Alpha, and Critical Values

Confidence Level	Alpha ($\alpha$)	Critical Value ($z_{\alpha/2}$)
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.575

Key Takeaways

• The sample proportion is the best point estimate for the population proportion.

• Confidence intervals provide a range of plausible values for the population parameter, with a specified level of confidence.

• The margin of error quantifies the uncertainty in the estimate.

• Sample size calculations are essential for planning studies to achieve desired precision.