Q: A sample of 50 students has a mean score of 80 and a standard deviation of 10. Find the 95% confidence interval for the population mean. (Use $$z = 1.96)$$

$$80 \pm 1.96 \times \frac{10}{\sqrt{50}}$$

Q: A researcher wants to estimate the proportion of students who own a laptop. What sample size is needed to estimate the proportion within $$\pm 0.05$$ with 95% confidence, assuming $$p = 0.5$$? (Use $$n = \frac{z^2$ p (1-p)}{E^2$}$$, $$z = 1.96$$, $$E = 0.05)$$

$$n = \frac{(1.96)^2$ \times 0.5 \times 0.5}{(0.05)^2$}$$

Question 1

A researcher collects the following data on the number of hours students study per week: 5, 8, 8, 10, 12. What is the median of this data set?

Accepted Answer

$$8$$

Question 2

Which of the following sampling methods is used when every member of the population has an equal chance of being selected, and selection is done purely by chance?

Accepted Answer

Simple random sampling

Question 3

A survey records the favorite color of 100 students. Is this data quantitative or qualitative?

Accepted Answer

Qualitative

Question 4

Which of the following is an example of a continuous variable?

Accepted Answer

Temperature measured in degrees Celsius

Question 5

According to the Range Rule of Thumb, what is the approximate standard deviation for the data set: 2, 4, 6, 8, 10?

Accepted Answer

$$\frac{10-2}{4} = 2$$

Question 6

If $$P(A) = 0.3$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.8$$

Question 7

A procedure involves flipping a coin 10 times and recording the number of heads. Is this a binomial experiment?

Accepted Answer

Yes, because each flip is independent and has two outcomes

Question 8

A fair die is rolled 6 times. What is the probability of getting exactly 2 sixes? (Use the binomial formula: $$P(X=k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$, where n=6$, k=2$, p$$=\frac{1}{6}$$)

Accepted Answer

$$\binom{6}{2} \left(\frac{1}{6}\$$right)^2$$ \left(\frac{5}{6}\$$right)^4$$

Question 9

The heights of adult males in a city are normally distributed with mean $$\mu = 70$$ inches and standard deviation $$\sigma = 3$$ inches. What is the probability that a randomly selected male is taller than 76 inches? (Use the standard normal table)

Accepted Answer

$$P(Z \u003e 2) \approx 0.0228$$

Question 10

Which statement best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution

Question 11

A sample of 50 students has a mean score of 80 and a standard deviation of 10. Find the 95% confidence interval for the population mean. (Use $$z = 1.96)$$

Accepted Answer

$$80 \pm 1.96 	imes \frac{10}{\sqrt{50}}$$

Question 12

Which of the following best describes a Type I error in hypothesis testing?

Accepted Answer

Rejecting a true null hypothesis

Question 13

A medical test for a disease has a 5% probability of a Type II error. What does this mean?

Accepted Answer

There is a 5% chance the test fails to detect the disease when it is present

Question 14

A hypothesis test yields a $$p-$$value of $$0.03. If $$the significance level $$\alpha is$$ $$0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis

Question 15

Given the following paired data: $$(1,2), (2,3), (3,5), (4,7)$$, calculate the linear correlation coefficient $$r.$$

Accepted Answer

$$r \approx 0.98$$

Question 16

If the calculated correlation coefficient $$r = 0.85$$ and the critical value $$r_c = 0.80$$, what can you conclude?

Accepted Answer

There is a significant linear correlation

Question 17

A linear regression analysis yields the equation $$y = 2x + 5. $$What is the predicted value of $$y$$ when $$x = 4$$?

Accepted Answer

$$y = 2 	imes 4 + 5 = 13$$

Question 18

A researcher wants to estimate the proportion of students who own a laptop. What sample size is needed to estimate the proportion within $$\pm 0.05$$ with 95% confidence, assuming $$p = 0.5$$? (Use $$n = \frac{z^2$ p (1-p)}{E^2$}$$, $$z = 1.96$$, $$E = 0.05)$$

Accepted Answer

$$n = \frac{(1.96)^2$ 	imes 0.5 	imes 0.5}{(0.05)^2$}$$

Question 19

A study finds $$r = 0.10$$ for the relationship between hours studied and exam score, with $$r_c = 0.30. $$What does this indicate?

Accepted Answer

There is no significant linear correlation

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