Exam Review: Sampling Distributions and Confidence Interval Estimation

Chapter 7: Sampling Distributions

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the distribution of sample means over repeated sampling from the same population. It is a foundational concept for statistical inference.

• Definition: The sampling distribution of the sample mean is the probability distribution of all possible sample means of a given sample size drawn from a population.

• Central Limit Theorem (CLT): The CLT states that, for sufficiently large sample sizes, the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution.

• Formula: $\mu_{\bar{x}} = \mu$ $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ where $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.

• Finding Probability: To find the probability of a sample mean, use the normal distribution (if applicable) and standardize using the z-score: $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

Example: If the population mean is 100 and the standard deviation is 15, for a sample size of 25, the standard deviation of the sample mean is $\frac{15}{\sqrt{25}} = 3$.

Sampling Distribution of Sample Proportion

The sampling distribution of the sample proportion describes the distribution of sample proportions over repeated samples.

• Definition: The sample proportion ($\hat{p}$) is the ratio of successes to the total sample size.

• Normal Approximation: The sampling distribution of $\hat{p}$ can be approximated by a normal distribution if $np \geq 5$ and $n(1-p) \geq 5$.

• Formula: $\mu_{\hat{p}} = p$ $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

Example: In a sample of 100 people, if 40% prefer product A, $\hat{p} = 0.4$ and $\sigma_{\hat{p}} = \sqrt{\frac{0.4 \times 0.6}{100}} = 0.049$.

Chapter 8: Confidence Interval Estimation

Constructing and Interpreting Confidence Intervals

Confidence intervals provide a range of values within which the population parameter is likely to fall, based on sample data.

• Definition: A confidence interval is an interval estimate, calculated from sample data, that is likely to contain the population parameter.

• Formula for Mean (Known $\sigma$): $\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$ where $z^*$ is the critical value from the standard normal distribution.

• Formula for Mean (Unknown $\sigma$): $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$ where $t^*$ is the critical value from the t-distribution and $s$ is the sample standard deviation.

• Formula for Proportion: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Margin of Error: The margin of error depends on the critical value and the standard error of the estimate.

• Sample Size: The sample size affects the width of the confidence interval; larger samples yield narrower intervals.

Example: For a sample mean of 50, $s = 10$, $n = 25$, and a 95% confidence level ($t^* \approx 2.064$): $50 \pm 2.064 \times \frac{10}{5} = 50 \pm 4.128$

Confidence Interval for Difference Between Means

When comparing two means, a confidence interval can be constructed for the difference between them.

• Formula: $(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

• Paired Data: For paired samples, use the mean and standard deviation of the differences.

• Independent Samples: For independent samples, use the formula above.

Example: Comparing test scores before and after a training program using paired data.

Confidence Interval for Population Proportion

Confidence intervals for population proportions estimate the true proportion of a characteristic in the population.

• Formula: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Sample Size Calculation: To achieve a desired margin of error $E$ at confidence level $z^*$: $n = \frac{z^{*2} p(1-p)}{E^2}$

Example: To estimate the proportion of customers satisfied with a service, use the formula above with sample data.

Interpreting Confidence Intervals

It is important to understand what a confidence interval means and what it does not mean.

• Correct Interpretation: A 95% confidence interval means that, in repeated sampling, 95% of such intervals will contain the true population parameter.

• Incorrect Interpretation: It does NOT mean there is a 95% probability that the parameter is in the interval for a single sample.

Example: "We are 95% confident that the true mean lies between 45 and 55."

Type of Confidence Interval	Formula	When to Use
Mean (Known $\sigma$)	$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$	Population standard deviation is known
Mean (Unknown $\sigma$)	$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$	Population standard deviation is unknown
Proportion	$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	Estimating population proportion
Difference Between Means	$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$	Comparing two independent samples

Additional info: These notes expand on the brief review points by providing definitions, formulas, and examples for each concept, ensuring a comprehensive understanding for exam preparation.