Q: Construct a 95% confidence interval for the true proportion of students who use tablets. Use $$z^* = 1.96. $$Which of the following is correct?

$$0.55 \pm 1.96 \times \sqrt{\frac{0.55 \times 0.45}{800}}$$

Q: A researcher wants to estimate a population proportion with 99% confidence and a margin of error of 0.02. No prior estimate is available. What formula should be used to determine the minimum sample size $$n$$?

$$n = \left( \frac{z^*}{E} \$$right)^2$$ \times 0.25$$

Q: A light bulb manufacturer tests 16 bulbs and finds a mean life of $$\bar{x} = 1150$$ hours with $$s = 40$$ hours. What is the correct formula for the 95% confidence interval for the mean life, using the $$t-$$distribution?

$$\bar{x} \pm t_{0.025,15} \times \frac{s}{\sqrt{n}}$$

Question 1

A random sample of 800 college students found that 440 of them use a tablet for taking notes. What is the point estimate $$\hat{p}$$ for the proportion of students who use a tablet?

Accepted Answer

$$\hat{p} = \frac{440}{800} = 0.55$$

Question 2

For the same sample, what is the value of $$\hat{q}$$?

Accepted Answer

$$\hat{q} = 1 - \hat{p} = 0.45$$

Question 3

Which of the following is a requirement for constructing a confidence interval for a population proportion?

Accepted Answer

$$np \geq 5$$ and $$nq \geq 5$$

Question 4

Construct a 95% confidence interval for the true proportion of students who use tablets. Use $$z^* = 1.96. $$Which of the following is correct?

Accepted Answer

$$0.55 \pm 1.96 	imes \sqrt{\frac{0.55 	imes 0.45}{800}}$$

Question 5

A university official claims that more than half of all students use tablets. If your 95% confidence interval for the proportion is (0.515, 0.585), what is the correct conclusion?

Accepted Answer

The interval supports the claim because it is entirely above 0.5.

Question 6

A researcher wants to estimate a population proportion with 99% confidence and a margin of error of 0.02. No prior estimate is available. What formula should be used to determine the minimum sample size $$n$$?

Accepted Answer

$$n = \left( \frac{z^*}{E} \$$right)^2$$ 	imes 0.25$$

Question 7

If a previous study suggested the population proportion is approximately 0.65, what value should be used for $$p(1-p) in $$the sample size formula?

Accepted Answer

$$0.65 	imes 0.35$$

Question 8

Does having a prior estimate for $$p$$ closer to 0 or 1 (away from 0.5) increase or decrease the required sample size for a given margin of error?

Accepted Answer

It decreases the required sample size.

Question 9

A light bulb manufacturer tests 16 bulbs and finds a mean life of $$\bar{x} = 1150$$ hours with $$s = 40$$ hours. What is the correct formula for the 95% confidence interval for the mean life, using the $$t-$$distribution?

Accepted Answer

$$\bar{x} \pm t_{0.025,15} 	imes \frac{s}{\sqrt{n}}$$

Question 10

A company claims more than 80% of its customers are satisfied. A test of 500 customers finds 390 are satisfied. What is the null hypothesis $$H_0$$ for this test?

Accepted Answer

$$H_0$$: p = 0.80$$

Question 11

For the same company, what is the correct formula for the $$z$$ test statistic for the sample proportion?

Accepted Answer

$$z = \frac{\hat{p} - p_0$}{\sqrt{\frac{p_0$ q_0$}{n}}}$$

Question 12

A gym claims its members lose more than 10 lbs in their first month. A sample of 12 members has $$\bar{x} = 12.5$$ lbs and $$s = 3.2$$ lbs. What is the $$t$$ test statistic for this sample?

Accepted Answer

$$t = \frac{12.5 - 10}{3.2/\sqrt{12}}$$

Question 13

A study compares the battery life of two smartphone brands. Brand A: $$n_1 = 30$, $$\bar{x}_1 = 14.5$$, s_1$ = 1.2. $$Brand B: $$n_2 = 30$, $$\bar{x}_2 = 13.8$$, s_2$ = 1.5. $$What is the correct formula for the test statistic $$t$$ assuming unequal variances?

Accepted Answer

$$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1$^2}{n_1$} + \frac{s_2$^2}{n_2$}}}$$

Question 14

A researcher wants to compare the proportion of people who prefer coffee in two cities. City 1: $$x_1 = 120$, n_1$ = 200. $$City 2: $$x_2 = 140$, n_2$ = 250. $$What is the null hypothesis $$H_0$$ for this test?

Accepted Answer

$$H_0$$: $$p_1$$ = $$p_2$$

Question 15

For the same coffee preference study, what is the correct formula for the 95% confidence interval for the difference in proportions $$p_1$$ - $$p_2$$?

Accepted Answer

$$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1 \hat{q}_1}{n_1$} + \frac{\hat{p}_2 \hat{q}_2}{n_2$}}$$

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