Chapters 8–12: Hypothesis Testing, Inferences from Two Samples, Correlation & Regression, Goodness-of-Fit, and ANOVA

Chapter 8: Hypothesis Testing

Testing a Population Mean

Hypothesis testing for a population mean involves determining whether sample data provide sufficient evidence to support or refute a claim about the population mean.

• Null Hypothesis (H0): A statement that the population mean equals a specific value.

• Alternative Hypothesis (H1): A statement that the population mean differs from the value in H0.

• Test Statistic: For known population standard deviation, use the z-test; for unknown, use the t-test.

• Significance Level (\( \alpha \)): The probability of rejecting H0 when it is true (commonly 0.05).

Formula (t-test):

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

• \( \bar{x} \): sample mean

• \( \mu_0 \): hypothesized population mean

• \( s \): sample standard deviation

• \( n \): sample size

Example: Testing if the average height of a plant species differs from 15 cm using a sample of 30 plants.

Testing a Population Proportion

Used to determine if the proportion of a population with a certain characteristic matches a claimed value.

• Test Statistic (z-test):

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$

• \( \hat{p} \): sample proportion

• \( p_0 \): hypothesized population proportion

• \( n \): sample size

Example: Testing if the proportion of defective items in a shipment exceeds 5%.

Testing a Population Standard Deviation or Variance

Used to test claims about the variability of a population.

• Test Statistic (Chi-Square):

$\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$

• \( s^2 \): sample variance

• \( \sigma_0^2 \): hypothesized population variance

• \( n \): sample size

Example: Testing if the variance in exam scores is greater than a specified value.

Chapter 9: Inferences from Two Samples

Difference Between Two Proportions

Used to compare the proportions of two independent populations.

• Confidence Interval:

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

• \( \hat{p}_1, \hat{p}_2 \): sample proportions

• \( n_1, n_2 \): sample sizes

Example: Comparing the proportion of smokers in two different cities.

Difference Between Two Means (Independent Samples)

Used to test if the means of two independent populations are equal.

• Test Statistic (Equal Variances):

$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

Where pooled standard deviation \( s_p \) is:

$s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}$

• \( \bar{x}_1, \bar{x}_2 \): sample means

• \( s_1, s_2 \): sample standard deviations

• \( n_1, n_2 \): sample sizes

Example: Testing if average test scores differ between two schools.

Matched Pairs (Dependent Samples)

Used when samples are paired or matched (e.g., before-and-after measurements).

• Test Statistic: Same as one-sample t-test, but applied to the differences.

$t = \frac{\bar{d}}{s_d / \sqrt{n}}$

• \( \bar{d} \): mean of the differences

• \( s_d \): standard deviation of the differences

• \( n \): number of pairs

Example: Comparing blood pressure before and after treatment in the same patients.

F Test for Two Variances

Used to compare the variances of two populations.

• Test Statistic:

$F = \frac{s_1^2}{s_2^2}$

• \( s_1^2 \): variance of sample 1

• \( s_2^2 \): variance of sample 2

Example: Testing if the variability in weights differs between two factories.

Chapter 10: Correlation and Regression

Linear Correlation

Measures the strength and direction of the linear relationship between two variables.

• Correlation Coefficient (r): Ranges from -1 to 1.

$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$

• Properties: r = 1 (perfect positive), r = -1 (perfect negative), r = 0 (no linear correlation).

Example: Analyzing the relationship between hours studied and exam scores.

Regression Equations and Predictions

Regression analysis estimates the relationship between variables and allows predictions.

• Regression Equation:

$\hat{y} = b_0 + b_1 x$

• \( b_1 = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)

• \( b_0 = \bar{y} - b_1 \bar{x} \)

Example: Predicting sales based on advertising expenditure.

Coefficient of Determination (R2)

Indicates the proportion of the variance in the dependent variable explained by the regression model.

• Interpretation: R2 = 0.85 means 85% of the variation is explained by the model.

Example: Comparing models to determine which best fits the data.

Chapter 11: Goodness-of-Fit Tests

Chi-Square Goodness-of-Fit Test

Used to determine if a sample matches a population with a specific distribution.

• Test Statistic:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

• \( O_i \): observed frequency

• \( E_i \): expected frequency

Example: Testing if a die is fair based on observed roll frequencies.

Chapter 12: Analysis of Variance (ANOVA)

One-Way ANOVA

Used to test if three or more population means are equal.

• Hypotheses:

  • H0: All population means are equal.

  • H1: At least one mean is different.

• Test Statistic (F):

$F = \frac{\text{Between-group variance}}{\text{Within-group variance}}$

• If F is significantly large, reject H0.

Example: Comparing average yields of three different crop varieties.

Summary Table: Key Tests and Their Purposes

Test	Purpose	Test Statistic
t-test	Test mean (one or two samples)	t
z-test	Test proportion	z
Chi-Square	Test variance or distribution fit	\( \chi^2 \)
F-test	Compare two variances or ANOVA	F
Correlation	Measure linear relationship	r
Regression	Predict values	\( \hat{y} = b_0 + b_1 x \)

Additional info: This guide covers the main objectives and statistical tests from Chapters 8–12, including hypothesis testing, inference from two samples, correlation and regression, goodness-of-fit, and ANOVA. For each test, ensure you understand the assumptions, how to calculate the test statistic, and how to interpret results.