Math Problem Statement

Professors collected data consisting of eye color and gender of statistics students. Among 921 male​ students, 360 had blue eyes. Among 1092 female​ students, 362 had blue eyes. Use a 00.05 significance level to test the claim that the proportions of blue eyes are the same for males and females. Complete parts​ (a) through​ (c) below. Consider the first sample to be the sample of males and the second sample to be the sample of females.

Question content area bottom

Part 1

a. Test the claim using a hypothesis test.

What are the null and alternative hypotheses for the hypothesis​ test?

A.

Upper H 0H0​: p 1p1equals=p 2p2

Upper H 1H1​: p 1p1less than<p 2p2

B.

Upper H 0H0​: p 1p1greater than or equals≥p 2p2

Upper H 1H1​: p 1p1not equals≠p 2p2

C.

Upper H 0H0​: p 1p1equals=p 2p2

Upper H 1H1​: p 1p1not equals≠p 2p2

D.

Upper H 0H0​: p 1p1less than or equals≤p 2p2

Upper H 1H1​: p 1p1not equals≠p 2p2

E.

Upper H 0H0​: p 1p1equals=p 2p2

Upper H 1H1​: p 1p1greater than>p 2p2

F.

Upper H 0H0​: p 1p1not equals≠p 2p2

Upper H 1H1​: p 1p1equals=p 2p2

Solution

For this hypothesis test, we are testing the claim that the proportions of blue eyes are the same for males and females. Therefore:

  • Null Hypothesis H0H_0: The proportion of males with blue eyes is equal to the proportion of females with blue eyes, p1=p2p_1 = p_2.
  • Alternative Hypothesis H1H_1: The proportion of males with blue eyes is not equal to the proportion of females with blue eyes, p1≠p2p_1 \neq p_2.

The correct answer is therefore:

C. H0H_0: p1=p2p_1 = p_2, H1H_1: p1≠p2p_1 \neq p_2

Would you like help with parts (b) and (c) or further clarification on hypothesis testing?

Here are five questions to further explore the topic:

  1. What is the formula for calculating the test statistic for a two-sample proportion test?
  2. How do we interpret a p-value in hypothesis testing?
  3. What is the significance level, and how does it affect hypothesis testing?
  4. Why is it important to check assumptions, such as sample size, when conducting a hypothesis test?
  5. How do we decide between a one-tailed and two-tailed test in hypothesis testing?

Tip: In hypothesis testing, always start by defining the null and alternative hypotheses clearly before proceeding with calculations. This helps ensure clarity throughout the process.

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Math Problem Analysis

Mathematical Concepts

Statistics
Hypothesis Testing
Proportion Testing

Formulas

Test statistic for two proportions: z = (p1 - p2) / sqrt(p(1 - p)(1/n1 + 1/n2))
Pooled proportion: p = (x1 + x2) / (n1 + n2)

Theorems

Central Limit Theorem
Law of Large Numbers

Suitable Grade Level

Grades 11-12, College Level

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