Q: For the financial analyst's claim (Question 1), what is the value of the test statistic $$z$$? Use $$p_0 = 0.60$, sample proportion $$\hat{p} = 0.45$$, and n = 100$.

$$z = \frac{0.45 - 0.60}{\sqrt{\frac{0.60 \times 0.40}{100}}} = -3.06$$

Q: For the fast-food chain's claim, what is the correct null and alternative hypothesis?

$$H_0$$: p = 0.25;\quad H_a: p \neq 0.25$$

Q: Calculate the $$z-$$statistic for the fast-food chain's claim using $$p_0 = 0.25$, $$\hat{p} = 0.20$$, and n = 120$.

$$z = \frac{0.20 - 0.25}{\sqrt{\frac{0.25 \times 0.75}{120}}} = -1.00$$

Q: What is the correct null and alternative hypothesis for comparing the two insurance policy claim proportions?

$$H_0$$: $$p_1$$ = $$p_2$$;\quad H_a: $$p_1$$ \neq $$p_2$$

Q: Calculate the standard error for the difference in proportions for the insurance company data, using the pooled proportion $$\hat{p} = 0.0922$$, $$n_1 = 500$, n_2$ = 400.$$

$$SE = \sqrt{0.0922 \times 0.9078 \left( \frac{1}{500} + \frac{1}{400} \right)} = 0.0202$$

Question 1

A financial analyst claims that less than 60% of all investors in a mutual fund prefer a hands-off, passive investing strategy. In a random sample of 100 investors, 45 prefer a hands-off approach. At a 1% significance level, what is the correct null and alternative hypothesis for this test?

Accepted Answer

$$H_0$$: p \geq 0.60;\quad H_a: p \u003c 0.60$$

Question 2

For the financial analyst's claim (Question 1), what is the value of the test statistic $$z$$? Use $$p_0 = 0.60$, sample proportion $$\hat{p} = 0.45$$, and n = 100$.

Accepted Answer

$$z = \frac{0.45 - 0.60}{\sqrt{\frac{0.60 	imes 0.40}{100}}} = -3.06$$

Question 3

Given the calculated $$z-$$value of $$-3.06$$ for the analyst's claim and a significance level of 1%, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$; there is statistically significant evidence that less than 60% prefer a hands-off strategy.

Question 4

A fast-food chain claims that 25% of their customers order fries. In a sample of 120 customers, 24 ordered fries. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{24}{120} = 0.20$$

Question 5

For the fast-food chain's claim, what is the correct null and alternative hypothesis?

Accepted Answer

$$H_0$$: p = 0.25;\quad H_a: p 
eq 0.25$$

Question 6

Calculate the $$z-$$statistic for the fast-food chain's claim using $$p_0 = 0.25$, $$\hat{p} = 0.20$$, and n = 120$.

Accepted Answer

$$z = \frac{0.20 - 0.25}{\sqrt{\frac{0.25 	imes 0.75}{120}}} = -1.00$$

Question 7

At a significance level of 20% ($$\alpha = 0.20$$), what is the critical value for a two-tailed $$z-$$test?

Accepted Answer

$$z_{\alpha/2} = \pm 1.2816$$

Question 8

Given the calculated $$z-$$value of $$-1.00$$ and the critical value $$\pm 1.2816$$ for the fast-food chain's claim, what is the correct conclusion?

Accepted Answer

Fail to reject $$H_0$$; there is not enough evidence against the chain's claim.

Question 9

An insurance company wants to test if the proportion of claims differs between standard and premium policy holders. For standard: $$n_1 = 500$, x_1$ = 45$$; for premium: $$n_2 = 400$, x_2$ = 38. $$What is the pooled sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{45 + 38}{500 + 400} = 0.0922$$

Question 10

What is the correct null and alternative hypothesis for comparing the two insurance policy claim proportions?

Accepted Answer

$$H_0$$: $$p_1$$ = $$p_2$$;\quad H_a: $$p_1$$ 
eq $$p_2$$

Question 11

Calculate the standard error for the difference in proportions for the insurance company data, using the pooled proportion $$\hat{p} = 0.0922$$, $$n_1 = 500$, n_2$ = 400.$$

Accepted Answer

$$SE = \sqrt{0.0922 	imes 0.9078 \left( \frac{1}{500} + \frac{1}{400} ight)} = 0.0202$$

Question 12

What is the value of the test statistic $$z$$ for the insurance company data? Use $$\hat{p}_1 = 0.09$$, $$\hat{p}_2 = 0.095$$, and $$SE = 0.0202.$$

Accepted Answer

$$z = \frac{0.09 - 0.095}{0.0202} = -0.2475$$

Question 13

At a 5% significance level, what is the critical value for a two-tailed $$z-$$test?

Accepted Answer

$$z_{\alpha/2} = \pm 1.96$$

Question 14

Given the calculated $$z-$$value of $$-0.2475$$ and the critical value $$\pm 1.96$$, what is the correct conclusion for the insurance company data?

Accepted Answer

Fail to reject $$H_0$$; there is no significant difference in claim proportions.

Question 15

If the insurance company incorrectly concludes that there is a difference in claim proportions when there is not, what type of error is this?

Accepted Answer

Type I error

Question 16

Which of the following best describes the meaning of a p-value in hypothesis testing?

Accepted Answer

The probability of observing a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true.

Hypothesis Testing for Proportions – STAT 101 Assignment Guidance

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