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

Ch. 13 – Sampling Distributions and Confidence Intervals for Proportions

Normal Sampling Distribution for One Sample Proportion

Understanding when the sampling distribution of a sample proportion is approximately normal is crucial for inference about population proportions.

• Conditions for Normality: The sampling distribution of the sample proportion \( \hat{p} \) is approximately normal if:

  • The sample is random.

  • The population is at least 10 times larger than the sample (10% condition).

  • Both \( np \geq 10 \) and \( n(1-p) \geq 10 \), where \( n \) is the sample size and \( p \) is the population proportion.

• Sampling Distribution: For a sample proportion \( \hat{p} \):

  • Mean: $\mu_{\hat{p}} = p$

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

• Example: If 60% of a population prefers a product (\( p = 0.6 \)), and a random sample of 100 is taken, the sampling distribution of \( \hat{p} \) is approximately normal with mean 0.6 and standard deviation $\sqrt{\frac{0.6 \times 0.4}{100}} = 0.049$.

Constructing Confidence Intervals for a Population Proportion

Confidence intervals estimate the true population proportion based on sample data.

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

• \( z^* \) is the critical value from the standard normal distribution for the desired confidence level (e.g., 1.96 for 95%).

• Interpretation: "We are 95% confident that the true population proportion lies within this interval."

• Example: In a sample of 200, 120 favor a policy. \( \hat{p} = 0.6 \). 95% CI: $0.6 \pm 1.96 \times \sqrt{\frac{0.6 \times 0.4}{200}} = 0.6 \pm 0.068$.

Determining Sample Size for a Desired Margin of Error

• Formula (with preliminary estimate \( \hat{p} \)): $n = \left( \frac{z^*}{E} \right)^2 \hat{p}(1-\hat{p})$

• Formula (no estimate): Use \( \hat{p} = 0.5 \) for maximum variability.

• Example: For a margin of error of 0.03 at 95% confidence, $n = \left( \frac{1.96}{0.03} \right)^2 \times 0.5 \times 0.5 = 1067$.

Ch. 14 – Confidence Intervals for Means

Normal Sampling Distribution for One Sample Mean

The sampling distribution of the sample mean \( \bar{x} \) is approximately normal under certain conditions.

• Conditions for Normality:

  • Random sample.

  • Population is normal, or sample size is large (Central Limit Theorem: \( n \geq 30 \)).

• Sampling Distribution:

  • Mean: $\mu_{\bar{x}} = \mu$

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

Constructing Confidence Intervals for a Population Mean

• Formula (when \( \sigma \) unknown): $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

• \( t^* \) is the critical value from the t-distribution with \( n-1 \) degrees of freedom.

• Example: Sample mean = 50, s = 10, n = 25, 95% CI: $50 \pm 2.064 \times \frac{10}{5} = 50 \pm 4.13$.

Margin of Error and Confidence Interval Length

• Margin of Error (E): The maximum expected difference between the true population parameter and a sample estimate.

• Length of Confidence Interval: $2 \times E$

• Increasing sample size decreases margin of error; higher confidence level increases margin of error.

Ch. 15 – Testing Hypotheses

Null and Alternative Hypotheses for One-Sample z or t Test

• Null Hypothesis (\( H_0 \)): Statement of no effect or no difference (e.g., \( H_0: \mu = \mu_0 \)).

• Alternative Hypothesis (\( H_a \)): Statement indicating the presence of an effect (e.g., \( H_a: \mu \neq \mu_0 \)).

• Choose one-tailed or two-tailed alternative based on research question.

p-value and Level of Significance

• p-value: Probability of observing a test statistic as extreme as, or more extreme than, the observed value under \( H_0 \).

• Level of Significance (\( \alpha \)): Threshold for rejecting \( H_0 \) (commonly 0.05).

• If p-value < \( \alpha \), reject \( H_0 \); otherwise, fail to reject \( H_0 \).

Performing Hypothesis Tests for Proportions and Means

• z-test for Proportion: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• t-test for Mean: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

• Compare test statistic to critical value or use p-value approach.

Ch. 17 – Comparing Groups

Confidence Intervals for the Difference in Two Population Parameters

• Difference in Proportions: $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

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

• Interpretation: The interval estimates the difference between two population parameters.

Hypothesis Tests for the Difference in Two Population Parameters

• Null Hypothesis: \( H_0: p_1 = p_2 \) or \( H_0: \mu_1 = \mu_2 \)

• Test Statistic: Use formulas above, compare to critical value or use p-value.

Ch. 18 – Paired Samples and Blocks

Identifying Paired Data

• Paired Data: Each observation in one sample is uniquely matched to an observation in the other sample (e.g., before-and-after measurements on the same subject).

• Paired data require different analysis than independent samples.

Confidence Intervals for Paired Sample Data

• Calculate the difference for each pair: \( d_i = x_{1i} - x_{2i} \).

• Construct a confidence interval for the mean difference \( \mu_d \): $\bar{d} \pm t^* \frac{s_d}{\sqrt{n}}$

Paired Sample t-Test

• Null Hypothesis: \( H_0: \mu_d = 0 \)

• Test Statistic: $t = \frac{\bar{d}}{\frac{s_d}{\sqrt{n}}}$

• Compare to t-distribution with \( n-1 \) degrees of freedom.

Test Format and Study Tips

• Test Format: 11 multiple-choice questions, 4 free-response questions.

• Resources Allowed: Calculator and one handwritten 3x5 inch index card (no worked examples).

• How to Study:

  • Complete review problems on Blackboard.

  • Review class notes and previous homework/quiz problems.

  • Prepare your index card with key formulas and concepts.