Sampling Distributions and Estimators: Concepts, Properties, and Applications

Sampling Distributions and Estimators

General Behavior of Sampling Distributions

Sampling distributions describe the behavior of statistics (such as proportions, means, and variances) calculated from repeated random samples of the same size drawn from a population. Understanding these distributions is fundamental for making statistical inferences about populations.

• Sample Proportions: When repeatedly sampling from a population, the distribution of sample proportions tends to be normally distributed as sample size increases.

• Mean of Sample Proportions: The mean of all sample proportions equals the population proportion.

Table: Sampling Distribution Properties

Statistic	Distribution Shape	Mean of Distribution
Sample Proportion ($ar{p}$)	Normal (for large n)	Population proportion ($p$)
Sample Mean ($ar{x}$)	Normal (for large n)	Population mean ($bc$)
Sample Variance ($s^2$)	Skewed	Population variance ($c^2$)

Sampling Distribution of Proportions

To illustrate, consider rolling a die 5 times and calculating the proportion of odd numbers (1, 3, or 5) in each sample. Repeating this process many times produces a distribution of sample proportions, which approaches normality as the number of samples increases.

• Notation:

  • $x$ = number of successes

  • $n$ = sample size

  • $N$ = population size

  • $ar{p} = x/n$ (sample proportion)

  • $p = x/N$ (population proportion)

• Example: Rolling a die 5 times, finding the proportion of odd numbers, and repeating the procedure demonstrates the normality of the sampling distribution of proportions.

Sampling Distribution of Means

When repeatedly sampling and calculating the mean of each sample, the distribution of sample means tends to be normal, especially as sample size increases. This is a key result of the Central Limit Theorem.

• Example: Roll a die 5 times, calculate the mean, and repeat the process many times. The distribution of these sample means will approximate a normal distribution.

Sampling Distribution of Variances

Unlike means and proportions, the sampling distribution of sample variances is typically skewed rather than normal.

• Example: Randomly select samples, calculate the variance for each, and observe the distribution's skewness.

Unbiased and Biased Estimators

Estimators are statistics used to estimate population parameters. They are classified as unbiased or biased based on whether their sampling distribution's mean equals the true population parameter.

• Unbiased Estimator: A statistic whose expected value equals the population parameter.

  • Examples: Sample proportion ($ar{p}$), sample mean ($ar{x}$), sample variance ($s^2$).

• Biased Estimator: A statistic whose expected value does not equal the population parameter.

  • Examples: Sample median, sample range, sample standard deviation ($s$).

• Important Note: The sample standard deviation ($s$) is a biased estimator of the population standard deviation ($c$), but the bias is small for large samples, so $s$ is often used to estimate $c$.

Notation for the Sampling Distribution of the Mean

When all possible simple random samples of size $n$ are drawn from a population with mean $bc$ and standard deviation $c$, the sampling distribution of the sample mean has:

• Mean: $bc_{ar{x}} = bc$

• Standard deviation (Standard Error): $c_{ar{x}} = c / cn$

Standard error of the mean is sometimes denoted as SEM.

Practical Rules for Real Applications Involving a Sample Mean

When applying normal distribution methods to sample means, certain requirements must be met:

• If the population is normal or $n > 30$:

  • Mean of all sample means: $bc_{ar{x}} = bc$

  • Standard deviation of all sample means: $c_{ar{x}} = c / cn$

  • Z-score conversion: $z = rac{ar{x} - bc}{c / cn}$

• If the population is not normal and $n c 30$:

  • The distribution of $ar{x}$ cannot be well-approximated by a normal distribution; normal methods do not apply.

Examples and Applications

Boeing 737 Airline Seats

Suppose American Airlines considers reducing seat width from 16.6 in. to 16.0 in. Adult male hip widths are normally distributed with mean 14.3 in. and standard deviation 0.9 in.

• Probability that a randomly selected male has hip width > 16.0 in.: $P(X > 16.0) = 1 - ext{NORM.DIST}(16, 14.3, 0.9, 1) = 0.0295$

• Probability that the mean hip width of 126 males > 16.0 in.: $P(ar{X} > 16.0) = 1 - ext{NORM.DIST}(16, 14.3, 0.9/cc126, 1) capprox 0$

• Interpretation: The result for individual seats (part a) is more relevant, as seats are occupied by individuals. Even a 3% probability means several passengers per flight may require special accommodation, making the seat width reduction impractical.

Body Temperatures

Assume population mean body temperature is 98.6b0F, standard deviation 0.62b0F. A sample of 106 subjects has mean 98.2b0F.

• Probability of sample mean $ar{x} c 98.2$: $P(ar{x} c 98.2) = ext{NORM.DIST}(98.2, 98.6, 0.62/cc106, 1) capprox 0$

• Interpretation: The extremely low probability suggests that the true population mean is likely lower than 98.6b0F; in reality, it appears closer to 98.2b0F.

Simulation in Excel

Simulations can be used to empirically demonstrate sampling distributions:

• Roll a die 5 times, calculate the mean.

• Repeat the procedure 1000 times to observe the distribution of sample means.

Introduction to Hypothesis Testing: Rare Event Rule

Statistical inference often relies on the Rare Event Rule: If, under a given assumption, the probability of an observed event is very small, and the event occurs, we conclude the assumption is probably incorrect.

• Application: Used to assess whether sample results are consistent with population assumptions.

Summary Table: Unbiased vs. Biased Estimators

Estimator	Unbiased?	Targets Parameter?
Sample Mean ($ar{x}$)	Yes	Population Mean ($bc$)
Sample Proportion ($ar{p}$)	Yes	Population Proportion ($p$)
Sample Variance ($s^2$)	Yes	Population Variance ($c^2$)
Sample Median	No	Population Median
Sample Range	No	Population Range
Sample Standard Deviation ($s$)	No	Population Standard Deviation ($c$)

Additional info: The Central Limit Theorem underpins the normality of sampling distributions for means and proportions, especially as sample size increases.