A newspaper story claims that more houses are purchased by singles now than singles 55 years ago. To test this claim, two studies were conducted on the buying habits of singles over the past 55 years. In the first study, 200200 house purchases in the current year were randomly selected and 8080 of those were made by singles. In the second study, again 200200 house purchases were randomly selected from 55 years ago and 5959 of those were made by single people. Test the newspaper’s claim using a 0.050.05 level of significance. Is there sufficient evidence to support the newspaper’s claim? Let singles now be Population 1 and let singles 55 years ago be Population 2.

Step 1 of 3 :  

State the null and alternative hypotheses for the test. Fill in the blank below.

To test the newspaper's claim that more houses are purchased by singles now than 55 years ago, we can set up the null and alternative hypotheses as follows:

Let \( p_1 \) be the proportion of house purchases made by singles **now** (Population 1), and let \( p_2 \) be the proportion of house purchases made by singles **55 years ago** (Population 2).

### Null Hypothesis (\( H_0 \)):
\[ H_0: p_1 \leq p_2 \]
This states that the proportion of houses purchased by singles now is **less than or equal to** the proportion of houses purchased by singles 55 years ago.

### Alternative Hypothesis (\( H_1 \)):
\[ H_1: p_1 > p_2 \]
This represents the newspaper's claim that the proportion of houses purchased by singles now is **greater than** the proportion 55 years ago.

These are appropriate for a **one-tailed test** with a 0.05 level of significance.

Would you like to proceed with the next step to calculate the test statistic and decision rule?

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Here are 5 related questions:
1. How do you calculate the sample proportions \( \hat{p}_1 \) and \( \hat{p}_2 \) for this test?
2. What is the formula for the test statistic in a hypothesis test comparing two proportions?
3. How do you interpret the p-value in the context of this hypothesis test?
4. What are Type I and Type II errors in hypothesis testing?
5. How would the conclusion change if the level of significance was 0.01 instead of 0.05?

**Tip**: When comparing two proportions, always ensure the sample sizes are large enough for the normal approximation to be valid.

A newspaper story claims that more houses are purchased by singles now than singles 55 years ago. To test this claim, two studies were conducted on the buying habits of singles over the past 55 years. In the first study, 200 house purchases in the current year were randomly selected, and 80 of those were made by singles. In the second study, again 200 house purchases were randomly selected from 55 years ago, and 59 of those were made by single people. Test the newspaper’s claim using a 0.05 level of significance. Is there sufficient evidence to support the newspaper’s claim? Let singles now be Population 1 and let singles 55 years ago be Population 2.

Test the claim that more houses are purchased by singles today than 55 years ago using a hypothesis test for two proportions. This analysis compares two sample proportions of house purchases by singles from 200 samples each, applying statistical concepts like the Z-test and pooled proportion.

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