Q: A random variable $$X$$ can take values 0, 1, or 2 with probabilities 0.2, 0.5, and 0.3 respectively. What is $$E[X]$$?

$$E[X] = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Q: A binomial experiment has $$n=5$$ trials and probability of success $$p=0.4. $$What is the probability of exactly 2 successes?

$$P(X=2) = \binom{5}{2} \times 0.4^2$ \times 0.6^3$ = 10 \times 0.16 \times 0.216 = 0.3456$$

Q: A company wants to know the probability that at least 1 out of 3 randomly selected products is defective, given the probability of a defective product is 0.1. What is the probability?

$$P(\text{at least 1 defective}) = 1 - P(\text{none defective}) = 1 - 0.9^3$ = 1 - 0.729 = 0.271$$

Question 1

A pharmaceutical company wants to estimate the average time it takes for a new drug to relieve pain. Which of the following best describes the population and sample in this scenario?

Accepted Answer

Population: All patients in the world; Sample: All patients in the company's clinical trial.

Question 2

Which sampling method is most likely to introduce bias if a researcher only surveys customers who visit a store between 9am and 11am?

Accepted Answer

Convenience sampling

Question 3

A company records the number of defective products each day for a month. Which type of data is this?

Accepted Answer

Quantitative discrete

Question 4

Which of the following is NOT a valid reason to use a histogram instead of a bar chart?

Accepted Answer

Histograms are used to compare categories.

Question 5

Given the following data set: 2, 4, 6, 8, 10. What is the mean?

Accepted Answer

Mean $$= \frac{2+4+6+8+10}{5} = 6$$

Question 6

A data set has the values: 3, 7, 9, 12, 15. What is the median?

Accepted Answer

Median $$= 9$$

Question 7

A company measures the heights (in cm) of 5 employees: 160, 165, 170, 175, 180. What is the standard deviation?

Accepted Answer

Standard deviation $$= \sqrt{\frac{(160-170)^2$ + (165-170)^2$ + (170-170)^2$ + (175-170)^2$ + (180-170)^2$}{4}} = \sqrt{62.5} \approx 7.91$$

Question 8

If a data set has a standard deviation of 0, what does this indicate about the data?

Accepted Answer

All values are identical.

Question 9

A student scored in the 80th percentile on a standardized test. What does this mean?

Accepted Answer

The student scored higher than 80% of the test takers.

Question 10

If the probability of rain tomorrow is 0.3, what is the probability that it will NOT rain?

Accepted Answer

Probability $$= 1 - 0.3 = 0.7$$

Question 11

A hospital records whether patients are male or female and whether they have diabetes. Which tool is best for analyzing the relationship between these two variables?

Accepted Answer

Contingency table

Question 12

A random variable $$X$$ can take values 0, 1, or 2 with probabilities 0.2, 0.5, and 0.3 respectively. What is $$E[X]$$?

Accepted Answer

$$E[X] = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 13

A binomial experiment has $$n=5$$ trials and probability of success $$p=0.4. $$What is the probability of exactly 2 successes?

Accepted Answer

$$P(X=2) = \binom{5}{2} 	imes 0.4^2$ 	imes 0.6^3$ = 10 	imes 0.16 	imes 0.216 = 0.3456$$

Question 14

A company wants to know the probability that at least 1 out of 3 randomly selected products is defective, given the probability of a defective product is 0.1. What is the probability?

Accepted Answer

$$P(	ext{at least 1 defective}) = 1 - P(	ext{none defective}) = 1 - 0.9^3$ = 1 - 0.729 = 0.271$$

Question 15

If a variable is normally distributed with mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what is the $$z-$$score for $$x = 70$$?

Accepted Answer

$$z = \frac{70 - 50}{10} = 2$$

Question 16

A sample of 100 students has a mean score of 75 and a population standard deviation of 10. What is the standard error of the mean?

Accepted Answer

Standard error $$= \frac{10}{\sqrt{100}} = 1$$

Question 17

According to the Central Limit Theorem, what happens to the sampling distribution of the sample mean as the sample size increases?

Accepted Answer

It approaches a normal distribution, regardless of the population's distribution.

Question 18

A survey finds that 60 out of 200 people prefer a new product. What is the sample proportion?

Accepted Answer

Sample proportion $$= \frac{60}{200} = 0.3$$

Question 19

A researcher wants to estimate the probability that a randomly selected person is left-handed. In a sample of 1000 people, 90 are left-handed. What is the estimated probability?

Accepted Answer

Estimated probability $$= \frac{90}{1000} = 0.09$$

MAT 137: Midterm Study Guide (Chapters 1–5)

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