Q: Given a probability distribution: $$x = 1, 2, 3$$ with $$P(x) = 0.2, 0.5, 0.3$$ respectively. What is the mean of the distribution?

$$1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 2.1$$

Q: A fair coin is flipped 10 times. What is the probability of getting exactly 7 heads?

$$\binom{10}{7} \times (0.5)^7$ \times (0.5)^3$

Q: A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the minimum score required to be in the top 10%?

$$100 + 1.28 \times 15 = 119.2$$

Question 1

A researcher collects data on the favorite colors of 100 students. Which level of measurement best describes this data?

Accepted Answer

Nominal

Question 2

Which of the following is an example of a parameter?

Accepted Answer

The mean income of all residents in a city

Question 3

Given the following frequency distribution for the number of books read by students in a month: 1 book (10 students), 2 books (20 students), 3 books (30 students), 4 books (40 students). What is the relative frequency for students who read 3 books?

Accepted Answer

$$\frac{30}{100}$$

Question 4

A histogram shows the distribution of exam scores for a class. Most scores are clustered on the left, with a long tail extending to the right. What type of skewness does this histogram indicate?

Accepted Answer

Right-skewed

Question 5

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

Accepted Answer

$$\frac{2+10}{2} = 6$$

Question 6

A sample of five test scores is: 70, 80, 90, 100, 110. What is the range?

Accepted Answer

$$110 - 70 = 40$$

Question 7

In a class of 30 students, two students are to be randomly selected for president and vice president. What is the probability that a specific student, Alex, is selected for either position?

Accepted Answer

$$\frac{2}{30}$$

Question 8

A restaurant has three generators. Each generator has a probability of $$0.2 of $$failing. What is the probability that at least one generator works?

Accepted Answer

$$1 - (0.2)^3$ = 1 - 0.008 = 0.992$$

Question 9

Given a probability distribution: $$x = 1, 2, 3$$ with $$P(x) = 0.2, 0.5, 0.3$$ respectively. What is the mean of the distribution?

Accepted Answer

$$1 	imes 0.2 + 2 	imes 0.5 + 3 	imes 0.3 = 2.1$$

Question 10

A fair coin is flipped 10 times. What is the probability of getting exactly 7 heads?

Accepted Answer

$$\binom{10}{7} 	imes (0.5)^7$ 	imes (0.5)^3$

Question 11

In a binomial experiment with $$n = 20$$ and $$p = 0.1$$, what is considered a significantly low number of successes?

Accepted Answer

A number of successes where $$P(X \leq k) \u003c 0.05$$

Question 12

A continuous uniform distribution ranges from 2 to 8. What is the probability that a randomly selected value is between 4 and 6?

Accepted Answer

$$\frac{6-4}{8-2} = \frac{2}{6} = \frac{1}{3}$$

Question 13

For a standard normal distribution, what is the probability that $$z is $$between $$-1$$ and $$1$$?

Accepted Answer

Approximately $$0.68$$

Question 14

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the minimum score required to be in the top 10%?

Accepted Answer

$$100 + 1.28 	imes 15 = 119.2$$

Question 15

A hospital has 500 patients. If the probability that a patient has a certain condition is $$0.2$$, how many patients are expected to have the condition?

Accepted Answer

$$500 	imes 0.2 = 100$$

Question 16

A population has a mean of $$50. $$Three samples of size $$n = 5$$ have means of $$48$$, $$52$$, and $$50. Is $$the sample mean a good estimator of the population mean?

Accepted Answer

Yes, because the average of the sample means is $$\frac{48+52+50}{3} = 50$$, which equals the population mean.

Question 17

Suppose the probability that a randomly selected person has blood type O is $$0.45. $$What is the probability that in a sample of 10 people, at least one has blood type O?

Accepted Answer

$$1 - (1 - 0.45$$)^{10}$$

Question 18

A normal distribution has mean $$\mu = 70$$ and standard deviation $$\sigma = 10. $$What is the probability that a randomly selected value is between $$60$$ and $$80$$?

Accepted Answer

Approximately $$0.68$$

Question 19

A sample of 36 students has a mean test score of $$75$$ and a population standard deviation of $$12. $$What is the probability that the sample mean is between $$72$$ and $$78$$? (Assume normality and use the central limit theorem.)

Accepted Answer

Find $$z$$ for $$72$$ and $$78$$: $$z = \frac{72-75}{12/\sqrt{36}} = -1.5$$, $$z = \frac{78-75}{12/\sqrt{36}} = 1.5. $$Probability is area between $$z = -1.5$$ and $$z = 1.5.$$

Introductory Statistics Midterm Study Guide

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