GQ Data Analysis - Hypothesis Testing

Hypothesis Testing

• Decision-making process for evaluating claims about a population.

• Involves defining the population, stating hypotheses, significance level, sample selection, data collection, calculations, and conclusion.

Statistical Tests Overview

• Two main tests for hypotheses concerning means: z-test and t-test.

• Traditional Method: Used since the formulation of hypothesis testing.

Examples of Hypotheses

• Situation A: Medication side effects on pulse rate:

  • $H0:\mu=82$

  • $H1:\mu\ne82$

• Situation B: Battery additive to increase life:

  • $H0:μ=$36$$

  • H1:μ>36

• Situation C: Insulation to lower heating bills:

  • $H_0:\mu=78$

  • H_1:\mu<78

Mathematical Symbols for Hypotheses

• Equal to: =

• Not equal to: $\ne$

• Greater than: >

• Less than: <

• Two-tailed test:

  • $H0:\mu=k$

  • $H1:μ≠k$

• Right-tailed test:

  • $H0:\mu=k$

  • H1:\mu>k

  Left-tailed test:

  • $H0:\mu=k$

  • H1:\mu<k

Common Phrases

• Phrases indicating 'greater than': Is greater than, is above, is higher than, etc.

• Phrases indicating 'equal to': Is equal to, is the same as, has not changed from, etc.

• Phrases indicating 'less than': Is less than, is below, is lower than, etc.

• Phrases indicating 'not equal to': Is not equal to, is different from, has changed from, etc.

Claims in Hypothesis Testing

• A claim can be the null or alternative hypothesis.

• Statistical evidence can support the claim if it is the alternative hypothesis.

• Statistical evidence can reject the claim if it is the null hypothesis.

Statistical Test

• Uses sample data to decide whether to reject the null hypothesis.

• Test value: Numerical value from a statistical test.

• Decision based on comparison of sample mean to population mean.

• Significant difference leads to rejection of the null hypothesis.

Possible Outcomes

• Type I error: Rejecting a true null hypothesis.

• Type II error: Failing to reject a false null hypothesis.

Hypothesis Testing and Jury Trial Analogy

• Null hypothesis: Defendant is innocent.

• Alternative hypothesis: Defendant is guilty.

• Type I error: Convicting an innocent defendant.

• Type II error: Acquitting a guilty defendant.

Level of Significance

• Maximum probability of committing a type I error, denoted by αα.

• αα values: 0.10, 0.05, and 0.01 are common.

• Probability of a type II error is ββ.

Critical Value

• Separates the critical region from the noncritical region; symbol is C.V.

• Critical region: Values indicating significant difference, leading to rejection of the null hypothesis.

• Noncritical region: Values indicating difference due to chance; do not reject the null hypothesis.

• Location depends on the inequality sign of the alternative hypothesis (right-tailed, left-tailed).

Steps in Finding Critical Values

• Draw the figure and indicate the appropriate area based on the type of test (left, right, or two-tailed).

• Use the z table to find the critical value(s) corresponding to αα or α/2α/2.

Steps for Solving Hypothesis-Testing Problems (z-test)

• Step 1: State the hypotheses and identify the claim.

• Step 2: Find the z critical value(s).

• Step 3: Compute the z value.

• Step 4: Make the decision to reject or not reject the null hypothesis.

• Step 5: Summarize the results.

Z-Test Example Problem

• Problem: A researcher claims that the average height of adult males in a certain city is 175 cm. A sample of 30 adult males is taken, and the sample mean height is found to be 173 cm with a standard deviation of 5 cm. Test the hypothesis at a significance level of 0.05.

• Step 1: State the hypotheses:

  • Null hypothesis $(H0):\mu=175$

  • Alternative hypothesis $(H1):\mu\ne175$

• Step 2: Find the z critical value(s):

  • For α=0.05 in a two-tailed test, the critical values are $z=\pm1.96$

• Step 3: Compute the z value:

  • $z=xˉ−μσn=173−175530=−255.477=−20.913=−2.19z=n​σ​xˉ−μ​=30​5​173−175​=5.4775​−2​=0.913−2​=−2.19$

• Step 4: Make the decision:

  • Since the calculated z value (-2.19) is less than -1.96, we reject the null hypothesis.

• Step 5: Summarize the results:

  • There is sufficient evidence to reject the claim that the average height of adult males in the city is 175 cm at a significance level of 0.05.