Chapter 14: Sampling Distribution Models – Study Notes

Chapter 14: Sampling Distribution Models

Population vs. Sample, Parameter vs. Statistic

Understanding the distinction between populations and samples, as well as parameters and statistics, is foundational in statistics. These concepts underpin the logic of statistical inference.

• Population: The entire collection of individuals or items that one wishes to study.

• Sample: A subset of individuals selected from the population, ideally at random.

• Parameter: A numerical summary describing a characteristic of the population (e.g., population mean μ, population proportion p). The value is fixed but usually unknown.

• Statistic: A numerical summary calculated from a sample (e.g., sample mean \( \bar{y} \), sample proportion \( \hat{p} \)). The value varies from sample to sample and is used to estimate the parameter.

• Sampling Variability: The natural variation in statistics from one sample to another. Thus, a statistic is a random variable.

Example: If you want to know the average height of all students at a university (population), you might measure the heights of 100 randomly selected students (sample). The true average height is a parameter; the average from your sample is a statistic.

Sampling Distribution of Proportions (Percentages)

Definition and Properties

The sampling distribution of proportions describes the distribution of sample proportions \( \hat{p} \) from all possible random samples of a given size n from a population with true proportion p.

• Population Proportion (p): The true proportion of individuals in the population with a certain characteristic. $0 < p < 1$

• Sample Proportion (\( \hat{p} \)): The proportion in a sample with the characteristic. $\hat{p} = \frac{\text{# individuals sampled who have the characteristic}}{\text{sample size } n}$

• Sampling Distribution: The distribution of \( \hat{p} \) over all possible samples of size n from the population.

• Mean of Sampling Distribution: $\mu(\hat{p}) = p$

• Standard Deviation of Sampling Distribution: $\sigma(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}$

• Shape: For sufficiently large samples, the sampling distribution of \( \hat{p} \) is approximately Normal (by the Central Limit Theorem for proportions).

Assumptions and Conditions for Normal Approximation

• The sample is randomly drawn from the population.

• Individual values in the sample are independent. (If sampling without replacement, the sample size should be no more than 10% of the population.)

• The sample size is large enough: $np \geq 10$ and $n(1-p) \geq 10$

Example Table: Sampling Distribution of Proportions

Parameter	Statistic	Mean of Sampling Distribution	Standard Deviation
Population proportion (p)	Sample proportion (\( \hat{p} \))	p	$\sqrt{\frac{p(1-p)}{n}}$

Example Application

• Suppose 13% of patients who had eye laser surgery experienced post-surgical complications. A random sample of 1000 patients is taken.

• Parameter: $p = 0.13$ (true proportion in population)

• Statistic: $\hat{p}$ (proportion in the sample)

• Mean of $\hat{p}$: $0.13$

• Standard deviation: $\sqrt{\frac{0.13 \times 0.87}{1000}} \approx 0.0106$

• For large $n$, $\hat{p} \sim N(0.13, 0.0106)$ approximately.

Sampling Distribution of Means

Definition and Properties

The sampling distribution of means describes the distribution of sample means \( \bar{y} \) from all possible random samples of a given size n from a population with mean μ and standard deviation σ.

• Population Mean (μ): The true mean of the population.

• Sample Mean (\( \bar{y} \)): The mean of a sample of size n. $\bar{y} = \frac{y_1 + y_2 + \cdots + y_n}{n}$

• Sampling Distribution: The distribution of \( \bar{y} \) over all possible samples of size n from the population.

• Mean of Sampling Distribution: $\mu(\bar{y}) = \mu$

• Standard Deviation of Sampling Distribution: $\sigma(\bar{y}) = \frac{\sigma}{\sqrt{n}}$

• Shape: For sufficiently large samples, the sampling distribution of \( \bar{y} \) is approximately Normal, regardless of the population distribution (Central Limit Theorem).

The Central Limit Theorem (CLT)

• If $y_1, y_2, \ldots, y_n$ are independent random variables from any population with mean $\mu$ and standard deviation $\sigma$, then for large $n$, the distribution of $\bar{y}$ is approximately Normal:

$\bar{y} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$

• The larger the sample size $n$, the smaller the standard deviation of $\bar{y}$, and the better the Normal approximation.

• If the population is Normal, $\bar{y}$ is exactly Normal for any $n$.

Assumptions and Conditions for CLT

• The sample is randomly drawn from the population.

• Individual values in the sample are independent (sample size no more than 10% of the population).

• The sample size is sufficiently large (rule of thumb: $n \geq 30$ for most distributions).

Example Table: Sampling Distribution of Means

Parameter	Statistic	Mean of Sampling Distribution	Standard Deviation
Population mean (μ)	Sample mean (\( \bar{y} \))	μ	$\frac{\sigma}{\sqrt{n}}$

Example Application

• A manufacturer claims battery life is Normally distributed with mean 54 months and standard deviation 6 months. A sample of 50 batteries is tested.

• Parameter: $\mu = 54$ months, $\sigma = 6$ months

• Statistic: $\bar{y}$ (mean of sample of 50 batteries)

• Mean of $\bar{y}$: $54$

• Standard deviation: $\frac{6}{\sqrt{50}} \approx 0.849$

• For large $n$, $\bar{y} \sim N(54, 0.849)$ approximately.

Comparing Sampling Distributions: Effect of Sample Size

As sample size increases:

• The center of the sampling distribution remains at the population parameter (mean or proportion).

• The spread (standard deviation) decreases, making estimates more precise.

• The shape becomes more Normal, even if the population distribution is not Normal (by CLT).

Summary Table: Effect of Sample Size on Sampling Distribution

Sample Size (n)	Spread (SD)	Shape
Small	Larger	May not be Normal
Large	Smaller	Approximately Normal

Practice Exercises

• Exercise 1: Estimating the proportion of patients with post-surgical complications using a sample of 1000 patients. Identify population, sample, parameter, and statistic. Describe the sampling distribution and calculate probabilities related to sample proportions.

• Exercise 2: Battery life: Given population mean and standard deviation, describe the sampling distribution of the sample mean for a sample of 50 batteries. Calculate probabilities and confidence intervals for the sample mean.

• Exercise 3: Cola bottle filling: Given a Normal model for bottle contents, calculate the probability that an individual bottle or the mean of a six-pack is below a certain threshold.

Additional info: These notes are based on lecture slides and include both conceptual explanations and worked examples. The tables and formulas are reconstructed for clarity and completeness.