Statistics Unit 3: Confidence Intervals and Hypothesis Testing (Chapters 9-11) – Study Guide

Vocabulary and Notation

Key Terms

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

• Confidence Interval: An interval estimate, calculated from the sample data, that is likely to contain the population parameter with a specified level of confidence.

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

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

• Critical Value: The value that defines the endpoints of the confidence interval, based on the desired confidence level (e.g., z\( \alpha/2 \) or t\( \alpha/2 \)).

• Student’s t-Distribution: A probability distribution used when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

• Bootstrapping: A resampling method used to estimate the sampling distribution of a statistic by sampling with replacement from the original data.

• Percentile Method Confidence Interval: A confidence interval constructed from the percentiles of the bootstrap distribution.

• Hypothesis: A statement about a population parameter. Includes the null hypothesis (\( H_0 \)) and alternative hypothesis (\( H_1 \)).

• Hypothesis Testing: A statistical method for testing a claim about a population parameter using sample data.

• Type I Error: Rejecting the null hypothesis when it is true (false positive).

• Type II Error: Failing to reject the null hypothesis when it is false (false negative).

• Level of Significance (\( \alpha \)): The probability of making a Type I error.

• P-value: The probability, under the null hypothesis, of obtaining a result equal to or more extreme than what was actually observed.

• Statistical Significance: When the observed effect is unlikely to have occurred by chance, as determined by the p-value.

• Practical Significance: When the observed effect is large enough to be meaningful in real-world terms.

• Independent Samples: Samples in which the observations in one sample are not related to those in the other.

• Dependent Samples (Matched-Pairs): Samples in which each observation in one sample can be paired with an observation in the other sample.

• Robust Test: A statistical test that is valid even when certain assumptions are violated.

• Randomization Test: A nonparametric method for hypothesis testing using random resampling.

Notation

• 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 true population proportion based on a sample proportion.

• Formula:

$\hat{p} \pm z_{\alpha/2} \cdot \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

• Assumptions:

  • Sample obtained by simple random sampling or randomized experiment.

  • \( n\hat{p}(1-\hat{p}) \geq 10 \)

  • Sampled values are independent (sample size < 5% of population size).

• Example: If \( \hat{p} = 0.6 \), \( n = 100 \), and 95% confidence (\( z_{0.025} = 1.96 \)), the interval is: $0.6 \pm 1.96 \cdot \sqrt{\dfrac{0.6 \times 0.4}{100}} = 0.6 \pm 0.096$

t Confidence Interval for Mean

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

• Formula:

$\bar{x} \pm t_{\alpha/2} \cdot \dfrac{s}{\sqrt{n}}$

• 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: For \( \bar{x} = 50 \), \( s = 10 \), \( n = 25 \), 95% confidence (\( t_{0.025,24} \approx 2.064 \)): $50 \pm 2.064 \cdot \dfrac{10}{5} = 50 \pm 4.128$

One Sample Hypothesis Tests

z Test for One Sample Proportion

Tests whether the population proportion equals a specified value.

• Test Statistic:

$z_0 = \dfrac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$

• 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: \( H_0: p = p_0 \), \( H_1: p > p_0 \)

• Assumptions:

  • Simple random sample or randomized experiment.

  • \( n p_0 (1-p_0) \geq 10 \)

  • Sampled values are independent (sample size < 5% of population).

t Test for Mean

Tests whether the population mean equals a specified value when the population standard deviation is unknown.

• Test Statistic:

$t_0 = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$

• Degrees of Freedom: \( df = n - 1 \)

• 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: \( H_0: \mu = \mu_0 \), \( H_1: \mu > \mu_0 \)

• Assumptions:

  • Simple random sample or randomized experiment.

  • No outliers; population is normal or \( n \geq 30 \).

  • Sampled values are independent.

Two Sample Hypothesis Tests

Two Sample z Test for Proportions

Compares the proportions of two independent groups.

• Test Statistic:

$z_0 = \dfrac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\hat{p}(1-\hat{p})\left(\dfrac{1}{n_1} + \dfrac{1}{n_2}\right)}}$

where \( \hat{p} = \dfrac{x_1 + x_2}{n_1 + n_2} \)

• 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: \( H_0: p_1 = p_2 \), \( H_1: p_1 > p_2 \)

• Assumptions:

  • Independent samples from simple random sampling or randomized experiment.

  • \( n\hat{p}(1-\hat{p}) \geq 10 \) for both samples.

  • Sample sizes < 5% of respective populations.

Two Sample t Test for Dependent Means (Matched Pairs)

Compares means from two related groups (e.g., before and after measurements).

• Test Statistic:

$t_0 = \dfrac{\bar{d}}{s_d/\sqrt{n}}$

• Degrees of Freedom: \( df = n - 1 \)

• 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: \( H_0: \mu_d = 0 \), \( H_1: \mu_d > 0 \)

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

Two Sample t Test for Independent Means (Unequal Variances)

Compares means from two independent groups, not assuming equal variances.

• Test Statistic:

$t_0 = \dfrac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}$

• Degrees of Freedom (Welch-Satterthwaite):

$df = \dfrac{\left(\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}\right)^2}{\dfrac{1}{n_1-1}\left(\dfrac{s_1^2}{n_1}\right)^2 + \dfrac{1}{n_2-1}\left(\dfrac{s_2^2}{n_2}\right)^2}$

• 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: \( H_0: \mu_1 = \mu_2 \), \( H_1: \mu_1 > \mu_2 \)

• Assumptions:

  • Independent samples from simple random sampling or randomized experiment.

  • Populations are normal or sample sizes are large (\( n_1, n_2 \geq 30 \)).

  • Sample sizes < 5% of respective populations.

Two Sample Confidence Intervals

Confidence Interval for Difference Between Two Proportions

• Formula:

$(\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2} \cdot \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Assumptions:

  • Independent samples from simple random sampling or randomized experiment.

  • \( n_1\hat{p}_1(1-\hat{p}_1) \geq 10 \) and \( n_2\hat{p}_2(1-\hat{p}_2) \geq 10 \)

  • Sample sizes < 5% of respective populations.

Confidence Interval for Mean of Differences (Paired Data)

• Formula:

$\bar{d} \pm t_{\alpha/2} \cdot \dfrac{s_d}{\sqrt{n}}$

• Degrees of Freedom: \( df = n - 1 \)

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

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

• Formula:

$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \cdot \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$

• Degrees of Freedom: See formula above (Welch-Satterthwaite).

• Assumptions:

  • Independent samples from simple random sampling or randomized experiment.

  • Populations are normal or sample sizes are large (\( n_1, n_2 \geq 30 \)).

  • Sample sizes < 5% of respective populations.

Sample Size Calculations

Sample Size Needed for Proportions

• With Prior Estimate \( \hat{p} \):

$n = \hat{p}(1-\hat{p}) \left( \dfrac{z_{\alpha/2}}{E} \right)^2$

• Without Prior Estimate:

$n = 0.25 \left( \dfrac{z_{\alpha/2}}{E} \right)^2$

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

Sample Size Needed for Means

• Formula:

$n = \left( \dfrac{z_{\alpha/2} \cdot s}{E} \right)^2$

• Where E is the desired margin of error.

Summary Table: Hypothesis Tests and Confidence Intervals

Test/Interval	Parameter	Statistic	Assumptions	Formula
One-sample z for proportion	p	\( \hat{p} \)	Random sample, \( n p_0 (1-p_0) \geq 10 \), independence	$z_0 = \dfrac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$
One-sample t for mean	\( \mu \)	\( \bar{x} \)	Random sample, normality or large n, independence	$t_0 = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$
Two-sample z for proportions	\( p_1 - p_2 \)	\( \hat{p}_1 - \hat{p}_2 \)	Independent random samples, \( n\hat{p}(1-\hat{p}) \geq 10 \)	$z_0 = \dfrac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(1/n_1 + 1/n_2)}}$
Two-sample t for means (independent, unequal variances)	\( \mu_1 - \mu_2 \)	\( \bar{x}_1 - \bar{x}_2 \)	Independent random samples, normality or large n	$t_0 = \dfrac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}$
Paired t for means	\( \mu_d \)	\( \bar{d} \)	Random sample of pairs, normality or large n	$t_0 = \dfrac{\bar{d}}{s_d/\sqrt{n}}$

Additional Info

• Statistical software (e.g., StatCrunch) can be used to perform all calculations and simulations described above.

• Bootstrap and randomization methods provide alternatives to traditional parametric inference, especially when assumptions are questionable.