Hypothesis Testing for Proportions – STAT 101 Assignment Guidance

Q1. Is there statistically significant evidence at the 1% significance level that less than 60% of investors in a mutual fund prefer a hands-off strategy?

Background

Topic: Hypothesis Testing for a Single Proportion

This question tests your ability to conduct a hypothesis test for a population proportion using sample data, and to interpret the result in the context of a claim.

Key Terms and Formulas

• Null hypothesis ($H_0$): $p = 0.60$

• Alternative hypothesis ($H_a$): $p < 0.60$

• Sample proportion: $\hat{p} = \frac{45}{100}$

• Test statistic (z): $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• Significance level: $\alpha = 0.01$

Step-by-Step Guidance

1. State the hypotheses: $H_0: p = 0.60$, $H_a: p < 0.60$.

2. Calculate the sample proportion: $\hat{p} = \frac{45}{100}$.

3. Compute the standard error: $SE = \sqrt{\frac{0.60 \times 0.40}{100}}$.

4. Set up the z-test statistic formula: $z = \frac{\hat{p} - 0.60}{SE}$.

5. Determine the critical value for a one-tailed test at $\alpha = 0.01$.

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Q2. At the 20% significance level, is there statistically significant evidence against the fast-food chain's claim that 25% of customers order fries?

Background

Topic: Hypothesis Testing for a Single Proportion

This question asks you to test a claim about a population proportion using sample data, and to decide if the evidence is strong enough to reject the claim at a relatively high significance level.

Key Terms and Formulas

• Null hypothesis ($H_0$): $p = 0.25$

• Alternative hypothesis ($H_a$): $p \neq 0.25$

• Sample proportion: $\hat{p} = \frac{24}{120}$

• Test statistic (z): $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• Significance level: $\alpha = 0.20$

Step-by-Step Guidance

1. State the hypotheses: $H_0: p = 0.25$, $H_a: p \neq 0.25$.

2. Calculate the sample proportion: $\hat{p} = \frac{24}{120}$.

3. Compute the standard error: $SE = \sqrt{\frac{0.25 \times 0.75}{120}}$.

4. Set up the z-test statistic formula: $z = \frac{\hat{p} - 0.25}{SE}$.

5. Determine the critical values for a two-tailed test at $\alpha = 0.20$.

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Q3. Do the proportions of claims differ between Standard and Premium auto insurance policies at the 5% significance level? What type of error could occur if the decision is incorrect?

Background

Topic: Hypothesis Testing for the Difference Between Two Proportions

This question tests your ability to compare two population proportions using sample data, and to understand the types of errors in hypothesis testing.

Key Terms and Formulas

• Null hypothesis ($H_0$): $p_1 = p_2$

• Alternative hypothesis ($H_a$): $p_1 \neq p_2$

• Sample proportions: $\hat{p}_1 = \frac{45}{500}$, $\hat{p}_2 = \frac{38}{400}$

• Pooled proportion: $\hat{p} = \frac{45 + 38}{500 + 400}$

• Standard error: $SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{500} + \frac{1}{400}\right)}$

• Test statistic: $z = \frac{\hat{p}_1 - \hat{p}_2}{SE}$

• Significance level: $\alpha = 0.05$

Step-by-Step Guidance

1. State the hypotheses: $H_0: p_1 = p_2$, $H_a: p_1 \neq p_2$.

2. Calculate the sample proportions: $\hat{p}_1 = \frac{45}{500}$, $\hat{p}_2 = \frac{38}{400}$.

3. Compute the pooled proportion: $\hat{p} = \frac{45 + 38}{500 + 400}$.

4. Calculate the standard error: $SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{500} + \frac{1}{400}\right)}$.

5. Set up the z-test statistic formula: $z = \frac{\hat{p}_1 - \hat{p}_2}{SE}$.

6. Consider the types of errors: Type I (rejecting a true null) and Type II (failing to reject a false null).

Try solving on your own before revealing the answer!