A study was conducted to determine the proportion of people who dream in black and white instead of color. Among

321
 
people over the age of​ 55,

79
 
dream in black and​ white, and among

286
 
people under the age of​ 25,

13
 
dream in black and white. Use a

0.05
 
significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. Complete parts​ (a) through​ (c) below.

Question content area bottom

Part 1

a. Test the claim using a hypothesis test.

Consider the first sample to be the sample of people over the age of 55 and the second sample to be the sample of people under the age of 25. What are the null and alternative hypotheses for the hypothesis​ test?

To test the claim that the proportion of people over the age of 55 who dream in black and white is greater than the proportion of people under the age of 25, we'll first set up the null and alternative hypotheses based on the information provided.

### Step 1: Set up the hypotheses

Let:
- \( p_1 \) be the proportion of people over the age of 55 who dream in black and white.
- \( p_2 \) be the proportion of people under the age of 25 who dream in black and white.

We want to test whether the proportion of people over 55 dreaming in black and white is **greater** than that of those under 25.

#### Null Hypothesis \((H_0)\):
- The null hypothesis claims there is no difference or that the proportion of people over 55 dreaming in black and white is **less than or equal** to the proportion of people under 25.
\[
H_0: p_1 \leq p_2
\]

#### Alternative Hypothesis \((H_a)\):
- The alternative hypothesis claims that the proportion of people over 55 dreaming in black and white is **greater** than the proportion of people under 25.
\[
H_a: p_1 > p_2
\]

This is a **one-tailed test** because the alternative hypothesis involves testing whether \( p_1 \) is **greater than** \( p_2 \).

---

Let me know if you'd like to proceed with calculations for the hypothesis test, including the test statistic and the conclusion, or if you have any other questions!

Here are some related questions for you:
1. How do you calculate the sample proportions for both groups?
2. What is the formula for the test statistic in a two-sample proportion test?
3. How do you determine the critical value for a one-tailed test at the 0.05 significance level?
4. What is the interpretation of the p-value in a hypothesis test?
5. How do we conclude whether to reject or fail to reject the null hypothesis?

**Tip:** When setting up hypotheses, always make sure the alternative hypothesis directly reflects the claim being tested.

A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 321 people over the age of 55, 79 dream in black and white, and among 286 people under the age of 25, 13 dream in black and white. Use a 0.05 significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. Complete parts​ (a) through​ (c) below.

Test the claim that a greater proportion of people over 55 dream in black and white compared to those under 25 using hypothesis testing at a 0.05 significance level. This problem involves calculating sample proportions, performing a two-sample Z-test, and interpreting the results. Suitable for AP Statistics or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation