Statistics Unit 3: Estimation and Hypothesis Testing (Chapters 9-11) Study Guide

Estimation and Hypothesis Testing

Vocabulary and Notation

This section introduces key terms and notation used in statistical inference, including estimation and hypothesis testing.

• Point Estimate: A single value used to estimate a population parameter (e.g., sample mean \( \bar{x} \) estimates population mean \( \mu \)).

• Confidence Interval: A range of values, derived from sample statistics, that is likely to contain the population parameter.

• Level of Confidence: The probability that the confidence interval contains the true parameter, usually expressed as \( (1-\alpha) \cdot 100\% \).

• Margin of Error: The maximum expected difference between the point estimate and the true parameter.

• Critical Value: The value that marks the boundary for the desired confidence level (e.g., \( z_{\alpha/2} \) or \( t_{\alpha/2} \)).

• Student’s t-Distribution: A probability distribution used when estimating population mean with unknown variance and small sample size.

• Bootstrapping: A resampling method for estimating the distribution of a statistic.

• Percentile Method Confidence Interval: Uses percentiles from bootstrap samples to construct confidence intervals.

• Hypothesis: A claim about a population parameter. Includes Null Hypothesis (H0) and Alternative Hypothesis (H1).

• Hypothesis Testing: Procedure to assess evidence against H0 in favor of H1.

• Type I Error: Rejecting H0 when it is true.

• Type II Error: Failing to reject H0 when H1 is true.

• P-value: Probability of observing data as extreme as the sample, assuming H0 is true.

• Statistical Significance: When the P-value is less than the significance level (\( \alpha \)), indicating evidence against H0.

• Practical Significance: Whether the result has real-world importance.

• Independent Samples: Samples with no relationship between observations.

• Dependent Samples (Matched Pairs): Samples where observations are paired or related.

• Robust Test: A test that remains valid under violations of assumptions.

• Randomization Test: Uses random resampling to assess significance.

Parameter	Symbol
Population Proportion	p
Sample Proportion	\( \hat{p} \)
Population Mean	\( \mu \)
Sample Mean	\( \bar{x} \)
Sample Standard Deviation	s

One Sample Confidence Intervals

Confidence Interval for One Sample Proportion

Used to estimate the population proportion based on a sample.

• Formula:

  • Lower bound: $\hat{p} - z_{\alpha/2} \cdot \sqrt{ \frac{ \hat{p}(1-\hat{p}) }{ n } }$

  • Upper bound: $\hat{p} + z_{\alpha/2} \cdot \sqrt{ \frac{ \hat{p}(1-\hat{p}) }{ n } }$

• Assumptions:

  • Sample obtained by simple random sampling or randomized experiment.

  • $n \hat{p}(1-\hat{p}) \geq 10$

  • Sampled values are independent; sample size is less than 5% of population.

• Example: If $\hat{p} = 0.4$, $n = 100$, and $z_{\alpha/2} = 1.96$ for 95% confidence, calculate bounds.

t Confidence Interval for Mean

Used to estimate the population mean when the population standard deviation is unknown.

• Formula:

  • Lower bound: $\bar{x} - t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

  • Upper bound: $\bar{x} + t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

• Assumptions:

  • Sample obtained by simple random sampling or randomized experiment.

  • No outliers; population is normally distributed or $n \geq 30$.

  • Sampled values are independent.

• Example: $\bar{x} = 50$, $s = 10$, $n = 25$, $t_{\alpha/2} = 2.064$ for 95% confidence.

One Sample Hypothesis Tests

z Test for One Sample Proportion

Tests whether the sample proportion differs from a hypothesized value.

• Hypotheses:

  • Two-tailed: $H_0: p = p_0$, $H_1: p \neq p_0$

  • Left-tailed: $H_0: p = p_0$, $H_1: p < p_0$

  • Right-tailed: $H_0: p = p_0$, $H_1: p > p_0$

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

• Assumptions: Same as confidence interval for proportion.

t Test for Mean

Tests whether the sample mean differs from a hypothesized value.

• Hypotheses:

  • Two-tailed: $H_0: \mu = \mu_0$, $H_1: \mu \neq \mu_0$

  • Left-tailed: $H_0: \mu = \mu_0$, $H_1: \mu < \mu_0$

  • Right-tailed: $H_0: \mu = \mu_0$, $H_1: \mu > \mu_0$

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

• Degrees of Freedom: $df = n - 1$

• Assumptions: Same as t confidence interval for mean.

Two Sample Hypothesis Tests

Two Sample z Test for Proportions

Tests whether two population proportions are equal.

• Hypotheses:

  • Two-tailed: $H_0: p_1 = p_2$, $H_1: p_1 \neq p_2$

  • Left-tailed: $H_0: p_1 = p_2$, $H_1: p_1 < p_2$

  • Right-tailed: $H_0: p_1 = p_2$, $H_1: p_1 > p_2$

• Test Statistic: $z_0 = \frac{ ( \hat{p}_1 - \hat{p}_2 ) - ( p_1 - p_2 ) }{ \sqrt{ \hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } }$ where $\hat{p} = \frac{ x_1 + x_2 }{ n_1 + n_2 }$

• Assumptions: Samples are independent, random, and $n \hat{p}(1-\hat{p}) \geq 10$ for each sample.

Two Sample t Test for Dependent Means (Matched Pairs)

Tests whether the mean difference in paired data is zero.

• Hypotheses:

  • Two-tailed: $H_0: \mu_d = 0$, $H_1: \mu_d \neq 0$

  • Left-tailed: $H_0: \mu_d = 0$, $H_1: \mu_d < 0$

  • Right-tailed: $H_0: \mu_d = 0$, $H_1: \mu_d > 0$

• Test Statistic: $t_0 = \frac{ \bar{d} }{ s_d / \sqrt{n} }$

• Degrees of Freedom: $df = n - 1$

• Assumptions: Same as one sample mean, applied to differences.

Two Sample t Test for Independent Means (Unequal Variances)

Tests whether two population means are equal, assuming unequal variances.

• Hypotheses:

  • Two-tailed: $H_0: \mu_1 = \mu_2$, $H_1: \mu_1 \neq \mu_2$

  • Left-tailed: $H_0: \mu_1 = \mu_2$, $H_1: \mu_1 < \mu_2$

  • Right-tailed: $H_0: \mu_1 = \mu_2$, $H_1: \mu_1 > \mu_2$

• Test Statistic: $t_0 = \frac{ ( \bar{x}_1 - \bar{x}_2 ) - ( \mu_1 - \mu_2 ) }{ \sqrt{ \frac{ s_1^2 }{ n_1 } + \frac{ s_2^2 }{ n_2 } } }$

• Degrees of Freedom: $df = \frac{ \left( \frac{ s_1^2 }{ n_1 } + \frac{ s_2^2 }{ n_2 } \right)^2 }{ \frac{1}{ n_1 - 1 } \cdot \left( \frac{ s_1 }{ n_1 } \right)^2 + \frac{1}{ n_2 - 1 } \cdot \left( \frac{ s_2 }{ n_2 } \right)^2 }$

• Assumptions: Samples are independent, random, populations are normal or sample sizes are large.

Two Sample Confidence Intervals

Confidence Interval for Difference Between Two Proportions

Estimates the difference between two population proportions.

• Formula:

  • Lower bound: $( \hat{p}_1 - \hat{p}_2 ) - z_{\alpha/2} \cdot \sqrt{ \frac{ \hat{p}_1 (1-\hat{p}_1) }{ n_1 } + \frac{ \hat{p}_2 (1-\hat{p}_2) }{ n_2 } }$

  • Upper bound: $( \hat{p}_1 - \hat{p}_2 ) + z_{\alpha/2} \cdot \sqrt{ \frac{ \hat{p}_1 (1-\hat{p}_1) }{ n_1 } + \frac{ \hat{p}_2 (1-\hat{p}_2) }{ n_2 } }$

• Assumptions: Samples are independent, random, and $n_1 \hat{p}_1 (1-\hat{p}_1) \geq 10$, $n_2 \hat{p}_2 (1-\hat{p}_2) \geq 10$.

Confidence Interval for Mean of Differences (Paired Data)

Estimates the mean difference in paired data.

• Formula:

  • Lower bound: $\bar{d} - t_{\alpha/2} \cdot \frac{ s_d }{ \sqrt{n} }$

  • Upper bound: $\bar{d} + t_{\alpha/2} \cdot \frac{ s_d }{ \sqrt{n} }$

• Degrees of Freedom: $df = n - 1$

• Assumptions: Same as one sample mean, applied to differences.

Confidence Interval for Difference Between Two Independent Means (Unequal Variances)

Estimates the difference between two population means.

• Formula:

  • Lower bound: $( \bar{x}_1 - \bar{x}_2 ) - t_{\alpha/2} \cdot \sqrt{ \frac{ s_1^2 }{ n_1 } + \frac{ s_2^2 }{ n_2 } }$

  • Upper bound: $( \bar{x}_1 - \bar{x}_2 ) + t_{\alpha/2} \cdot \sqrt{ \frac{ s_1^2 }{ n_1 } + \frac{ s_2^2 }{ n_2 } }$

• Degrees of Freedom: $df = \frac{ \left( \frac{ s_1^2 }{ n_1 } + \frac{ s_2^2 }{ n_2 } \right)^2 }{ \frac{1}{ n_1 - 1 } \cdot \left( \frac{ s_1 }{ n_1 } \right)^2 + \frac{1}{ n_2 - 1 } \cdot \left( \frac{ s_2 }{ n_2 } \right)^2 }$

• Assumptions: Samples are independent, random, populations are normal or sample sizes are large.

Sample Size Calculations

Sample Size Needed for Proportions

Calculates the minimum sample size required to estimate a proportion with a specified margin of error.

• When prior estimate $\hat{p}$ is available: $n = \hat{p}(1-\hat{p}) \left( \frac{ z_{\alpha/2} }{ E } \right)^2$

• When no prior estimate: $n = 0.25 \left( \frac{ z_{\alpha/2} }{ E } \right)^2$

• Where $E$ is the desired margin of error (as a decimal).

Sample Size Needed for Means

Calculates the minimum sample size required to estimate a mean with a specified margin of error.

• Formula: $n = \left( \frac{ z_{\alpha/2} \cdot s }{ E } \right)^2$

• Where $E$ is the desired margin of error.

Summary Table: Hypothesis Tests and Confidence Intervals

Test/Interval	Parameter	Statistic	Assumptions
One Sample z for Proportion	p	$z_0 = \frac{ \hat{p} - p_0 }{ \sqrt{ \frac{ p_0 (1-p_0) }{ n } } }$	Random sample, independence, $n p_0 (1-p_0) \geq 10$
One Sample t for Mean	\( \mu \)	$t_0 = \frac{ \bar{x} - \mu_0 }{ s / \sqrt{n} }$	Random sample, normality or $n \geq 30$, independence
Two Sample z for Proportions	$p_1 - p_2$	$z_0 = \frac{ ( \hat{p}_1 - \hat{p}_2 ) }{ \sqrt{ \hat{p}(1-\hat{p}) ( \frac{1}{n_1} + \frac{1}{n_2} ) } }$	Random, independent samples, $n \hat{p}(1-\hat{p}) \geq 10$
Two Sample t for Means (Independent)	$\mu_1 - \mu_2$	$t_0 = \frac{ ( \bar{x}_1 - \bar{x}_2 ) }{ \sqrt{ \frac{ s_1^2 }{ n_1 } + \frac{ s_2^2 }{ n_2 } } }$	Random, independent, normality or large $n$
Paired t for Means	$\mu_d$	$t_0 = \frac{ \bar{d} }{ s_d / \sqrt{n} }$	Random, paired, normality or large $n$

Additional info: Bootstrapping and randomization tests are modern alternatives for inference, especially when assumptions are questionable. StatCrunch and similar software can be used for calculations and simulations.