Estimating Parameters and Determining Sample Sizes

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Estimating Parameters and Determining Sample Sizes

Estimating a Population Proportion

Estimating a population proportion involves using sample data to infer the true proportion of a characteristic in a population. The sample proportion is the best point estimate for the population proportion, and confidence intervals provide a range of plausible values for the population parameter.

Point Estimate: A single value used to approximate a population parameter. For proportions, the sample proportion \( \hat{p} \) is the best point estimate of the population proportion \( p \).
Example: If 43% of 1487 surveyed adults have a Facebook page, the point estimate for the population proportion is \( \hat{p} = 0.43 \).

Confidence Intervals for a Population Proportion

A confidence interval (CI) is a range of values used to estimate the true value of a population parameter. Common confidence levels are 90%, 95%, and 99%.

Confidence Level: The probability (1 − α) that the CI contains the population parameter if the process is repeated many times. For example, a 95% confidence level means that 95% of such intervals will contain the true parameter.
Correct Interpretation: "We are 95% confident that the interval from 0.405 to 0.455 contains the true value of the population proportion \( p \)."
Process Success Rate: Over many samples, the proportion of intervals that contain the true parameter equals the confidence level.

Confidence intervals from 20 different samples, with one not containing the true p

Critical Values and the Standard Normal Distribution

Critical values are z-scores that separate unlikely sample statistics from likely ones, based on the chosen confidence level. The value \( z_{\alpha/2} \) is used to construct confidence intervals.

Standard normal distribution showing critical values at alpha/2 in each tail Normal distribution with confidence level and alpha/2 regions marked

Common Critical Values: For a 95% confidence level, \( z_{\alpha/2} = 1.96 \). Other common values are shown in the table below.

Confidence level	α	Critical Value, \( z_{\alpha/2} \)
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.575

Table of confidence levels, alpha, and critical values

Margin of Error for Proportions

The margin of error (E) quantifies the maximum likely difference between the sample proportion and the true population proportion at a given confidence level.

Formula:

where \( \hat{q} = 1 - \hat{p} \), \( n \) is the sample size, and \( z_{\alpha/2} \) is the critical value.

Formula for confidence interval and margin of error for proportions

Conditions for Constructing a Confidence Interval for a Proportion

The sample must be a simple random sample.
The binomial distribution conditions are satisfied (fixed number of independent trials, two outcomes, constant probability).
There are at least 5 successes and 5 failures in the sample.

Determining Sample Size for Estimating a Population Proportion

To estimate a population proportion with a specified margin of error and confidence level, the required sample size can be calculated using the following formulas:

If an estimate \( \hat{p} \) is known:

If no estimate is known:

Formulas for sample size determination for proportions

Estimating a Population Mean

Estimating a Population Mean (σ Not Known)

When the population standard deviation (σ) is unknown, the Student t distribution is used to construct confidence intervals for the population mean (μ). The t distribution accounts for the extra variability in small samples.

Degrees of Freedom (df): The number of independent values that can vary in the sample, calculated as \( df = n - 1 \).
Margin of Error (E):

Confidence Interval:

Properties of the t Distribution:
- Different for each sample size (n).
- Symmetric and bell-shaped, but wider than the normal distribution for small n.
- Mean of t = 0; standard deviation > 1 and depends on n.
- As n increases, the t distribution approaches the normal distribution.

Comparison of t distributions for n=3, n=12, and standard normal distribution

Estimating a Population Mean (σ Known)

If the population standard deviation (σ) is known and the population is normally distributed or the sample size is large (n > 30), the normal (z) distribution is used.

Margin of Error (E):

Confidence Interval:

Choosing the Appropriate Distribution

Use the normal (z) distribution: σ known and population is normal or n > 30.
Use the t distribution: σ unknown and population is normal or n > 30.
Use nonparametric methods or bootstrapping: Population is not normal and n ≤ 30.

Additional info: These methods are foundational for inferential statistics, allowing researchers to make evidence-based conclusions about population parameters from sample data. Proper interpretation and application of confidence intervals and sample size calculations are essential for valid statistical inference.