Q: Given a probability distribution: x = 1, P(x) = 0.2; x = 2, P(x) = 0.5; x = 3, P(x) = 0.3. 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 tossed 10 times. What is the probability of getting exactly 7 heads?

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

Question 1

Which of the following best describes the difference between a parameter and a statistic?

Accepted Answer

A parameter is a value calculated from a population, while a statistic is a value calculated from a sample.

Question 2

A researcher records the blood types (A, B, AB, O) of 100 patients. What is the level of measurement for this data?

Accepted Answer

Nominal

Question 3

Given the following frequency distribution for the number of books read by students in a month: 0 books: 5 students, 1 book: 10 students, 2 books: 15 students, 3 books: 20 students. What is the relative frequency for students who read 2 books?

Accepted Answer

$$\frac{15}{50}$$

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 is the shape of this distribution?

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 weights (in kg) is: 50, 52, 54, 56, 58. What is the range?

Accepted Answer

$$58 - 50 = 8$$

Question 7

In a city, 60% of restaurants are fast food and 40% are sit-down. If 30% of all restaurants are both fast food and sit-down, what is the probability that a randomly selected restaurant is either fast food or sit-down?

Accepted Answer

$$0.60 + 0.40 - 0.30 = 0.70$$

Question 8

A committee has two positions to fill from a group of 10 students. What is the probability that the same student is selected for both positions if selection is without replacement?

Accepted Answer

$$0$$

Question 9

A generator has a 0.2 probability of failing. If there are three generators, what is the probability that at least one generator works?

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

A fair coin is tossed 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 12

In a binomial experiment with $$n = 20$$ and $$p = 0.1$$, is getting 0 successes significantly low?

Accepted Answer

Yes, because $$P(X \leq 0) \u003c 0.05$$

Question 13

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 14

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

Accepted Answer

$$0.682$$

Question 15

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the probability that a randomly selected value is less than 85?

Accepted Answer

$$P(x \u003c 85) = P(z \u003c \frac{85-100}{15}) = P(z \u003c -1) = 0.1587$$

Question 16

In a normal distribution with $$\mu = 50$$ and $$\sigma = 10$$, what is the minimum value required to be in the top 10%?

Accepted Answer

$$x = \mu + z \sigma$$,\ z = 1.28\ \Rightarrow\ x = 50 + 1.28 	imes 10 = 62.8$

Question 17

A population has mean $$\mu = 80. $$Three samples of size $$n = 4$$ have means 78, 82, and 80. Is the sample mean a good estimator of the population mean?

Accepted Answer

Yes, because the average of sample means $$\frac{78+82+80}{3} = 80$$ matches the population mean.

Question 18

A normal distribution has $$\mu = 70$$ and $$\sigma = 8. $$What is the probability that a randomly selected value is between 62 and 78?

Accepted Answer

$$P(62 \u003c x \u003c 78) = P(z \u003c 1) - P(z \u003c -1) = 0.8413 - 0.1587 = 0.6826$$

Question 19

A normal distribution has $$\mu = 60$$ and $$\sigma = 5. If 95$$% of students pass, what is the minimum score required to pass?

Accepted Answer

$$x = \mu + z \sigma$$,\ z = 1.645\ \Rightarrow\ x = 60 + 1.645 	imes 5 = 68.225$

Question 20

A normal distribution has $$\mu = 100$$ and $$\sigma = 20. $$What is the probability that a sample mean from $$n = 16$$ falls between 90 and 110?

Accepted Answer

$$P(90 \u003c \bar{x} \u003c 110) = P(z \u003c \frac{110-100}{5}) - P(z \u003c \frac{90-100}{5}) = P(z \u003c 2) - P(z \u003c -2) = 0.9772 - 0.0228 = 0.9544$$

Introductory Statistics Midterm Study Guide

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