Q: Suppose the probability of a defective item from a factory is $$p = 0.02. If 50 $$items are selected at random, what is the expected number of defective items?

$$E(X) = np = 50 \times 0.02 = 1$$

Q: A fair die is rolled 60 times. What is the expected value and standard deviation of the number of times a '6' appears? Use the binomial distribution.

$$E(X) = np = 60 \times \frac{1}{6} = 10$$, $$\sigma = \sqrt{np(1-p)} = \sqrt{60 \times \frac{1}{6} \times \frac{5}{6}} \approx 2.89$$

Question 1

Which of the following best describes the Central Limit Theorem (CLT) and its importance in inferential statistics?

Accepted Answer

The CLT states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 2

A pharmaceutical company wants to test if a new drug is more effective than the current standard. They randomly assign 100 patients to the new drug and 100 to the standard drug. Which statistical test is most appropriate to compare the proportions of patients who recover in each group?

Accepted Answer

A two-proportion z-test

Question 3

Suppose the probability of a defective item from a factory is $$p = 0.02. If 50 $$items are selected at random, what is the expected number of defective items?

Accepted Answer

$$E(X) = np = 50 	imes 0.02 = 1$$

Question 4

A researcher collects data on the heights (in cm) of 200 university students. Which measure would best describe the central tendency if the data is skewed to the right?

Accepted Answer

The median

Question 5

Which of the following is NOT an assumption for performing a one-sample t-test for the mean?

Accepted Answer

The population standard deviation is known.

Question 6

A fair die is rolled 60 times. What is the expected value and standard deviation of the number of times a '6' appears? Use the binomial distribution.

Accepted Answer

$$E(X) = np = 60 	imes \frac{1}{6} = 10$$, $$\sigma = \sqrt{np(1-p)} = \sqrt{60 	imes \frac{1}{6} 	imes \frac{5}{6}} \approx 2.89$$

Question 7

A 95% confidence interval for a population mean is calculated as $$(52.3, 57.7). $$Which of the following statements is correct?

Accepted Answer

If we took many samples, about 95% of the calculated intervals would contain the population mean.

Question 8

Which of the following best describes the purpose of a chi-square test for independence?

Accepted Answer

To determine if two categorical variables are associated in a population.

Question 9

A hospital administrator wants to know if the average wait time in the emergency room has changed from last year's mean of 45 minutes. She collects a random sample of 40 wait times and finds a sample mean of 48 minutes with a sample standard deviation of 12 minutes. What is the value of the test statistic for a two-sided hypothesis test?

Accepted Answer

$$t = \frac{48 - 45}{12/\sqrt{40}} = \frac{3}{1.897} \approx 1.58$$

Question 10

Which of the following sampling methods is most likely to produce a representative sample of a college's student body?

Accepted Answer

Simple random sampling from the college's student registry

Question 11

A regression analysis yields the equation $$\hat{y} = 2.5 + 0.8x. $$What does the slope coefficient 0.8 represent in this context?

Accepted Answer

For each unit increase in $$x$$, the predicted value of $$y$$ increases by 0.8 units.

Question 12

Which of the following is a correct interpretation of a p-value in hypothesis testing?

Accepted Answer

The probability of observing data as extreme as, or more extreme than, the sample result, assuming the null hypothesis is true.

Question 13

A study finds a correlation coefficient $$r = -0.85$$ between hours of exercise per week and resting heart rate. Which of the following is the best interpretation?

Accepted Answer

There is a strong negative linear relationship between hours of exercise and resting heart rate.

Question 14

A quality control manager wants to estimate the proportion of defective products in a large shipment. She takes a random sample of 200 products and finds 12 are defective. Construct a 95% confidence interval for the proportion of defective products. (Use $$z^* = 1.96)$$

Accepted Answer

$$\hat{p} = \frac{12}{200} = 0.06$$, SE $$= \sqrt{\frac{0.06 	imes 0.94}{200}} \approx 0.0168$$, CI: $$0.06 \pm 1.96 	imes 0.0168 = (0.027, 0.093)$$

Question 15

Which of the following scenarios best illustrates the use of a matched pairs t-test?

Accepted Answer

Comparing the mean blood pressure before and after treatment in the same group of patients.

Question 16

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the probability that a randomly selected value is greater than 130? (Use the standard normal table.)

Accepted Answer

First, $$z = \frac{130 - 100}{15} = 2$$. $$P(Z \u003e 2) = 1 - 0.9772 = 0.0228$$

Question 17

Which of the following is a key difference between categorical and numerical data?

Accepted Answer

Numerical data can be measured and ordered, while categorical data represent groups or categories.

Question 18

A researcher wants to test if the proportion of left-handed students at Douglas College is different from the national average of 10%. She surveys 150 students and finds 21 are left-handed. What is the value of the test statistic for a one-proportion z-test?

Accepted Answer

$$z = \frac{0.14 - 0.10}{\sqrt{0.10 	imes 0.90 / 150}} = \frac{0.04}{0.0245} \approx 1.63$$

Question 19

Which of the following best describes the meaning of statistical significance in hypothesis testing?

Accepted Answer

The observed result is unlikely to have occurred by random chance alone, given the null hypothesis is true.

MATH 1160 – Introduction to Statistics: Course Structure and Key Topics

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