Sampling Distributions, Confidence Intervals, and Hypothesis Testing: Study Guide

Sampling Distributions

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion describes the distribution of sample proportions from repeated random samples of a population. It is used to estimate population proportions and calculate probabilities.

• Mean of the sampling distribution: $\mu_{\hat{p}} = p$

• Standard deviation (standard error): $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

• Normality condition: The distribution is approximately normal if $np(1-p) \geq 10$.

• Application: Use the mean and standard deviation in the normal calculator to compute probabilities or percentiles for sample proportions.

• Unusual probabilities: Probabilities less than 0.05 are considered unusual.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the distribution of sample means from repeated random samples. It is used to estimate population means and calculate probabilities.

• Mean of the sampling distribution: $\mu_{\bar{x}} = \mu$

• Standard deviation (standard error): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Normality condition: The distribution is approximately normal if the population is normal or if $n \geq 30$ (Central Limit Theorem).

• Application: Use the mean and standard deviation in the normal calculator to compute probabilities or percentiles for sample means.

Conditions for Proportion and Mean Problems

Proportion Problems	Mean Problems
Variable is Qualitative	Variable is Quantitative
Sample is random or unbiased	Sample is random or unbiased
Sample size ≤ 5% of population	Sample size ≤ 5% of population
$np(1-p) \geq 10$ for normality	Population normal or $n \geq 30$

Confidence Intervals

Confidence Interval for a Population Proportion

A confidence interval estimates the range in which the true population proportion is likely to fall, based on sample data.

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

• Critical z-score ($z^*$): The z-score corresponding to the desired confidence level (e.g., 1.96 for 95%).

• Sample proportion ($\hat{p}$): Point estimate of the population proportion.

• Sample size determination: $n = \frac{z^{*2} \hat{p}(1-\hat{p})}{E^2}$, where $E$ is the desired margin of error.

• StatCrunch application: Summarize qualitative variables to obtain $\hat{p}$ and calculate the confidence interval.

Confidence Interval for a Population Mean

A confidence interval estimates the range in which the true population mean is likely to fall, based on sample data.

• Formula: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

• Critical t-score ($t^*$): The t-score corresponding to the desired confidence level and degrees of freedom ($n-1$).

• Sample mean ($\bar{x}$): Point estimate of the population mean.

• Sample standard deviation ($s$): Used in the formula for the margin of error.

• Sample size determination: $n = \frac{t^{*2} s^2}{E^2}$, where $E$ is the desired margin of error.

• StatCrunch application: Summarize quantitative variables to obtain $\bar{x}$ and $s$ and calculate the confidence interval.

Interpreting Confidence Intervals

• Confidence level: $(1-\alpha) \times 100\%$ confidence means that, in repeated sampling, the interval will contain the true parameter $(1-\alpha) \times 100\%$ of the time.

• Width of interval: Higher confidence level or smaller sample size increases the width; larger sample size decreases the width.

• Using intervals for hypothesis testing: If the hypothesized value is outside the interval, reject the null hypothesis.

• Point estimate and margin of error: Point estimate = midpoint of interval; margin of error = half the width.

Hypothesis Testing

Setting Up Hypotheses

Hypothesis testing involves making a claim about a population parameter and using sample data to test that claim.

• Null hypothesis ($H_0$): Statement of no effect or no difference.

• Alternative hypothesis ($H_a$): Statement of effect or difference.

• Symbols: Use $p$ for proportions, $\mu$ for means.

• Test direction: Right-tailed ($>$), left-tailed ($<$), or two-tailed ($\neq$) tests.

Hypothesis Test for Proportion

• Test statistic: $z_0 = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• P-value: Probability of observing a test statistic as extreme as $z_0$ under $H_0$.

• Decision rule: Compare p-value to significance level ($\alpha$); reject $H_0$ if p-value $< \alpha$.

• Conclusion: State whether there is enough evidence to support $H_a$.

• Distribution: Draw the z-distribution and label the p-value area.

Hypothesis Test for Mean

• Test statistic: $t_0 = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

• P-value: Probability of observing a test statistic as extreme as $t_0$ under $H_0$.

• Decision rule: Compare p-value to significance level ($\alpha$); reject $H_0$ if p-value $< \alpha$.

• Conclusion: State whether there is enough evidence to support $H_a$.

• Distribution: Draw the t-distribution and label the p-value area.

Example: Hypothesis Test for Proportion

• Suppose $H_0: p = 0.5$, $H_a: p \neq 0.5$.

• Sample: $\hat{p} = 0.55$, $n = 100$.

• Calculate $z_0$:

• $z_0 = \frac{0.55 - 0.5}{\sqrt{\frac{0.5 \times 0.5}{100}}} = \frac{0.05}{0.05} = 1$

• Find p-value using normal distribution.

• Compare p-value to $\alpha$ (e.g., 0.05) and make a decision.

Example: Confidence Interval for Mean

• Sample: $\bar{x} = 20$, $s = 4$, $n = 25$, 95% confidence.

• Critical t-score ($t^*$) for 24 degrees of freedom ≈ 2.064.

• Margin of error: $2.064 \times \frac{4}{\sqrt{25}} = 2.064 \times 0.8 = 1.651$

• Confidence interval: $20 \pm 1.651 = (18.349, 21.651)$

Additional info: StatCrunch is a statistical software tool used to summarize data and perform calculations for confidence intervals and hypothesis tests.