Hypothesis Testing in Statistics

Research Questions and Data Analysis

• Use a SAMPLE to draw conclusions about a TARGET POPULATION.
• Differentiate between population and sample using appropriate notation.

Population Notation Refresher

• Sample notation is used for samples; Population notation for populations.
• Sample Statistics estimate Population Parameters:
  • Mean: $\bar{y}$ (Sample Mean) vs. $<br />\nu$ (Population Mean)
  • Median: $ilde{y}$ (Sample Median) vs. $ilde{<br />\nu}$ (Population Median)
  • Standard Deviation: $s$ (Sample SD) vs. $<br />\nu$ (Population SD)
  • Variance: $s^2$ (Sample Variance) vs. $<br />\nu^2$ (Population Variance)

Hypothesis Testing Overview

• Involves a series of formal steps to answer business and research questions.
• Example question: "Is the Outlier Oven pizza shop selling on average 200 pizzas per day or following an incentive scheme?"
  • Sample mean found: 204.6.
  • Population variance is an unknown variable.

Null Hypothesis (H0)

• A statement that a population parameter has a known value.
• In this case: $H_0: <br />\nu = 200$ (mean daily sales is equal to 200).

Alternative Hypothesis (H1)

• Opposite of the null hypothesis; it claims the population parameter is different from the null.
• For the pizza question: $H_1: <br />\nu <br />\neq 200$.

Null Distribution

• The distribution of the test statistic is known under the null hypothesis.
• The data must be normally distributed with a known variance for accurate results.

Test Statistic

• Used to determine how far the sample statistic is from the null hypothesis.
• The test statistic observed in the example: $z = 3$ (for the sample mean).

p-Value

• Measures the probability of obtaining a result as extreme or more extreme than the observed, under the null hypothesis.
• p-value represents the shaded area under the curve in relation to the test statistic.

Significant Result

• At a significance level of 5% (B1 = 0.05):
  • If p < 0.05, reject $H<em>0$ (evidence against $H</em>0$).

Non-Significant Result

• If p > 0.05, fail to reject $H<em>0$ (no evidence against $H</em>0$).

Conclusion for Outlier Oven

• Tested: $H<em>0: \nu = 200$ against $H</em>1: <br />\nu <br />\neq 200$.
• Test statistic: $t = 4.28$; p-value = 0 (less than 0.05).
• Conclusion: Evidence suggests Outlier Oven sells significantly more than 200 pizzas.

Hypothesis Testing Steps (H.A.T.P.D.C.)

1. Hypotheses: State $H<em>0$ and $H</em>1$.
2. Assumptions: Check assumptions of the test.
3. Test Statistic: Calculate the test statistic.
4. p-value: Obtain p-value.
5. Decision: Reject or not reject $H_0$ based on the p-value.
6. Conclusion: Conclude regarding the original research question.

Statistical Tests Overview

One Sample z-Test

• Use when:
  • One numeric variable exists.
  • Population standard deviation is known.

One Sample t-Test

• Criteria:
  • Single numeric variable.
  • Population standard deviation is unknown.
  • Compare sample mean to hypothesized mean.

Assumptions for One Sample t-Test

• Scores are numeric.
• Observations are independent.
• Appropriately normally distributed (n >= 25 can apply CLT, normality assumed otherwise).

Conclusion for One Sample t-Test

• p-value > 0.05: Do not reject $H_0$.
• p-value < 0.05: Reject $H_0$.
• Mention the business question in conclusions.

Two Sample t-Test

• For independent samples with two means:
  • Null: $H<em>0: \nu</em>1 = <br />\nu_2$.
  • Alternative: $H<em>1: \nu</em>1 <br />\neq <br />\nu_2$.

Assumptions for Two Sample t-Test

• Observations are independent.
• Observations are moderately normally distributed.
• Handle equal/unequal variance differently if applicable.

Conclusion for Two Sample t-Test

• Same as above: p-value conditioning (less than or greater than 0.05) with reference to the business question.

Paired Sample t-Test

• Calculate differences for samples of matched pairs:
  • Null: $H<em>0: \nu</em>d = 0$.
  • Alternative: $H<em>1: \nu</em>d <br />\neq 0$.

Chi-Square Goodness of Fit Test

• Used for categorical variables.
• State null and alternative hypotheses regarding proportions.

Example of Chi-Square Goodness of Fit

• Assess patterns in sick leave.
• Use proportions to evaluate conditions in the null hypothesis.
• Test validity with expected counts.