Probability Questions

PriviteraStats2 5.E.007.

Given two outcomes, A and B, state their relationship as a conditional probability using a “given” statement. (Select all that apply.)

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the probability of A, given that B occurs

the probability of B, given that A occurs

the probability given that neither A nor B occur

the probability given that A or B occurs the probability given that A and B occur

PriviteraStats2 5.E.009.

State the formula for calculating the following.

(a) the mean of a probability distribution

μ =

	xp

μ =

x − p n − 1

μ =

	(x − μ)²p

μ =

σ²

(b) the variance of a probability distribution

σ² =

	(x − μ)²p

σ² =

	xp

σ² =

x − p n − 1

σ² =

σ =

	(x − μ)²p

σ =

x − p n − 1

σ =

σ²

σ =

	xp

PriviteraStats2 5.E.011.

A hypothetical population consists of eight individuals ages 13, 16, 17, 18, 19, 26, 28, and 30 years. (Enter your answers to three decimal places.)

(a) What is the probability that a person in this population is a teenager?

(b) What is the probability of selecting a participant who is at least 20 years old?

(c) What is the probability of selecting a participant older than 30?

PriviteraStats2 5.E.013.

Researchers often enter a lot of data into statistical software programs. The probability of making zero to two errors per 1,000 keystrokes is 0.57, and the probability of making three to five errors per 1,000 keystrokes is 0.22. Find the probabilities (per 1,000 keystrokes) associated with each of the following.

(a) at most two errors

(b) at least three errors

(c) at most five errors

(d) more than five errors

PriviteraStats2 5.E.019.

The probability that a student in college will “experiment” with drugs is

p = 0.29.

The probability that a college student will not experiment with drugs is

q = 0.71.

(a) What type of relationship do these probabilities have?

independent complementary multiplicative supplementary

(b) A student heads off to college. What is the probability that the student experiments with drugs or does not experiment with drugs?

PriviteraStats2 5.E.021.

The probability that a student passes a class is

p(P) = 0.57.

The probability that a student studied for a class is

p(S) = 0.52.

The probability that a student passes a class given that he or she studied for the class is

p(P / S) = 0.74.

What is the probability that a student studied for the class, given that he or she passed the class

(p(S / P))?

Hint: Use Bayes’ theorem. (Round your answer to two decimal places.)

p(S / P) =

PriviteraStats2 5.E.025.

The relative frequency distribution of the number of phobias reported by a hypothetical sample of 500 college students is given as follows.

0–2	0.46
3–5	0.25
6–8	0.15
9–11	0.09
12–14	0.05

(a) What is the probability that a college student expresses fewer than three phobias?

(b) What is the probability that a college student expresses more than eight phobias?

(c) What is the probability that a college student has between 3 and 11 phobias?

PriviteraStats2 5.E.029.

Suppose a researcher is interested in the number of good versus bad dreams that students have during final exam week. The researcher states that

p = 0.46

that a student will have a bad dream during final exam week.

(a) What type of probability distribution is appropriate for these data?

bimodal distribution negatively skewed distribution positively skewed distribution binomial distribution

(b) Assuming complementary outcomes, what is the probability (q) that a student will have a good dream?
q =

(c) If a professor has 50 students in his class, then how many students should he expect to have bad dreams during final exam week?
students

PriviteraStats2 5.E.031.

In 2002, the Centers for Disease Control and Prevention (CDC) reported that 8% of women married for the first time by their 18th birthday, 25% married by their 20th birthday, and 76% married by their 30th birthday. Based on these data, what is the probability that in a family with two daughters, the first and second daughter will be married by the following ages? (Enter your answers to four decimal places.)

(a) 18 years of age

(b) 20 years of age

(c) 30 years of age

10.

PriviteraStats2 5.E.033.

Bar-eli, Azar, Ritov, Keidar-Levin, and Schein (2007) analyzed 286 penalty kicks among professional soccer players. They recorded the direction that the ball was kicked (left, center, right) and the direction that the goalie went to block the kick (left, center, right). The table shows the results of their study.

	Jump direction
Left	Center	Right	Total
Kick direction	Left	54	1	37	92
Center	41	10	31	82
Right	46	7	59	112
Total	141	18	127	286

Based on this table, answer the following questions. (Round your answers to two decimal places.)

(a) What is the probability that the ball was kicked to the left?

(b) What is the probability that the ball was kicked left and the goalie jumped left?

(c) What is the probability that the goalie jumped to the left, given that the ball was kicked to the left? Note: This is now a conditional probability statement.

11.

PriviteraStats2 6.E.009.

What type of distribution does the binomial distribution approximate?

a bimodal distribution a normal distribution a positively skewed distribution a negatively skewed distribution

12.

PriviteraStats2 6.E.011.

Using the unit normal table, find the proportion under the standard normal curve that lies to the right of the following values. (Round your answers to four decimal places.)

(a)

z = 2.00

(b)

z = −1.45

(c)

z = 0

(d)

z = −2.70

(e)

z = 1.96

13.

PriviteraStats2 6.E.013.

Using the unit normal table, find the proportion under the standard normal curve that lies between the following values. (Round your answers to four decimal places.)

(a) the mean and

z = 1.96

(b) the mean and

z = 0

(c)

z = −1.80 and z = 1.80

(d)

z = −0.40 and z = −0.30

(e)

z = 1.00 and z = 2.00

14.

PriviteraStats2 6.E.015.

An athletics coach states that the distribution of player run times (in seconds) for a 100-meter dash are normally distributed with a mean equal to 15.00 and a standard deviation equal to 0.2 seconds. What percentage of players on the team runs the 100-meter dash in faster than 15.30 seconds? (Round your answer to two decimal places.)
%

You may need to use the appropriate table in Appendix B to answer this question.

15.

PriviteraStats2 6.E.017.

A sample of final exam scores is normally distributed with a mean equal to 21 and a variance equal to 16.

Part (a)

What percentage of scores are between 17 and 25? (Round your answer to two decimal places.)
%

Part (b)

What raw score is the cutoff for the top 10% of scores? (Round your answer to one decimal place.)

Part (c)

What is the proportion below 15? (Round your answer to four decimal places.)

Part (d)

What is the probability of a score less than 27? (Round your answer to four decimal places.)

You may need to use the appropriate table in Appendix B to answer this question.

16.

PriviteraStats2 6.E.019.

A set of data is normally distributed with a mean of 3.5 and a standard deviation of 0.6. State whether the first area is bigger, the second area is bigger, or the two areas are equal in each of the following situations for these data.

Part (a)

the area above the mean or the area below the mean

The first area is bigger. The second area is bigger. The two areas are equal.

Part (b)

the area between 2.9 and 4.1 or the area between 3.5 and 4.7

The first area is bigger. The second area is bigger. The two areas are equal.

Part (c)

the area between the mean and 3.5 or the area above 5.3

The first area is bigger. The second area is bigger. The two areas are equal.

Part (d)

the area below 3.6 or the area above 3.4

The first area is bigger. The second area is bigger. The two areas are equal.

Part (e)

the area between 4.1 and 4.7 or the area between 2.9 and 3.5

The first area is bigger. The second area is bigger. The two areas are equal.

You may need to use the appropriate table in Appendix B to answer this question.

17.

PriviteraStats2 6.E.021.

A normal distribution has a mean equal to 53. What is the standard deviation of this normal distribution if 2.5% of the proportion under the curve lies to the right of

x = 64.76?

(Round your answer to two decimal places.)

You may need to use the appropriate table in Appendix B to answer this question.

18.

PriviteraStats2 6.E.023.

A normal distribution has a standard deviation equal to 35. What is the mean of this normal distribution if the probability of scoring above

x = 215

is 0.0228? (Round your answer to one decimal place.)

You may need to use the appropriate table in Appendix B to answer this question.

19.

PriviteraStats2 6.E.025.

According to national data, about 11% of American college students earn a graduate degree. Using this estimate, what is the probability that exactly 24 undergraduates in a random sample of 200 students will earn a college degree? Hint: Use the normal approximation to the binomial distribution, where

p = 0.11

and

q = 0.89.

(Round your answer to four decimal places.)

You may need to use the appropriate table in Appendix B to answer this question.

20.

PriviteraStats2 6.E.027.

McCabe, Ricciardelli, and James (2007) recruited 107 men and 151 women to complete a series of surveys pertaining to factors such as body image and body satisfaction. Using the Body Image Satisfaction scale, where higher scores indicate greater satisfaction, they found that men scored

19.10 ± 4.55 (M ± SD),

whereas women scored

16.84 ± 5.66 (M ± SD)

on this scale. Assuming these data are normally distributed, answer the following questions.

(a) What is the z-score for 19.10 in the sample of data for men?
z =

(b) What is the z-score for 16.84 in the sample of data for women?
z =

(c) What can you say about these two z-scores?

The first z-score is greater than the second because the sum of the mean and standard deviation for men is greater than that for women. The second z-score is greater than the first because 5.66, the standard deviation for men, is greater than 4.55, the standard deviation for women. The z-scores are the same because both scores are at the mean in their respective distributions. The first z-score is greater than the second because 19.10, the mean for men, is greater than 16.84, the mean for women.

21.

PriviteraStats2 6.E.029.

Montoya (2007) asked 56 men and 82 women to rate 21 different body parts on a scale of 1 (no opinion) to 5 (very desirable). They found that men and women rated the eyes similarly, with an average rating of about

3.77 ± 1.23 (M ± SD).

Assuming these data are normally distributed, answer the following questions. (Round your answers to two decimal places.)

(a) What percentage of participants rated the eyes at least a 5 (very desirable)?
%

(b) What percentage rated the eyes at most a 1 (no opinion)?
%

You may need to use the appropriate table in Appendix B to answer this question.

22.

PriviteraStats2 7.E.011.

What does the standard error measure?

Standard error measures the distance that sample mean values can be expected to deviate from the value of the population mean. Standard error measures the distance from the maximum value of the sample means to the minimum value of the sample means. Standard error measures the dispersion or deviation of sample values from the sample mean. Standard error measures the most frequently occurring amount that the sample means deviate from the population mean.

23.

PriviteraStats2 7.E.013.

A statistics instructor wants to measure the effectiveness of his teaching skills in a class of 69 students

(N = 69).

He selects students by waiting at the door to the classroom prior to his lecture and pulling aside every third student to give him or her a questionnaire.

(a) Is this sample design an example of random sampling? Explain.

Yes, because each student has an equal chance of being selected, and each student is replaced before selecting another student. No, because each student is not selected at random, and each student is not replaced before selecting another student.

(b) Assuming that all students attend his class that day, how many students will he select to complete the questionnaire?
students

24.

PriviteraStats2 7.E.015.

A clinical psychologist is the primary therapist for 13 patients. She randomly selects a sample of 3 patients to be in her study. How many different samples of this size can be selected from this population of 13 patients using theoretical sampling and experimental sampling?

(a) theoretical sampling
samples

(b) experimental sampling
samples

25.

PriviteraStats2 7.E.017.

Using the experimental sampling strategy, how many samples of size 4

(n = 4)

can be drawn from the following population sizes?

(a)

N = 6

samples

(b)

N = 7

samples

(c)

N = 8

samples

(d)

N = 9

samples

26.

PriviteraStats2 7.E.019.

Using the central limit theorem, what is the distribution of sample means when the population distribution is the following?

Part (a)

rectangular

negatively skewed evenly distributed positively skewed uniformly distributed normally distributed

Part (b)

normally distributed

negatively skewed positively skewed normally distributed evenly distributed uniformly distributed

Part (c)

positively skewed

evenly distributed uniformly distributed negatively skewed positively skewed normally distributed

Part (d)

nonmodal

negatively skewed positively skewed uniformly distributed normally distributed evenly distributed

Part (e)

multimodal

evenly distributed uniformly distributed positively skewed normally distributed negatively skewed

Part (f)

negatively skewed

uniformly distributed negatively skewed normally distributed evenly distributed positively skewed

27.

PriviteraStats2 7.E.021.

Using the skewed distribution rule, what is the distribution of sample variances when the population distribution is the following?

Part (a)

rectangular

negatively skewed normally distributed evenly distributed uniformally distributed positively skewed

Part (b)

normally distributed

normally distributed uniformally distributed positively skewed evenly distributed negatively skewed

Part (c)

positively skewed

negatively skewed evenly distributed normally distributed uniformally distributed positively skewed

Part (d)

nonmodal

negatively skewed normally distributed positively skewed evenly distributed uniformally distributed

Part (e)

multimodal

normally distributed positively skewed uniformally distributed negatively skewed evenly distributed

Part (f)

negatively skewed

negatively skewed uniformally distributed normally distributed positively skewed evenly distributed

28.

PriviteraStats2 7.E.023.

A population is normally distributed with a mean of 55 and a standard deviation of 14.

(a) What is the mean of the sampling distribution (μ_M) for this population?
μ_M =

(b) If a sample of 49 participants is selected from this population, what is the standard error of the mean (σ_M)?
σ_M =

(c) Sketch the shape of this distribution with

M ± 3 SEM.

29.

PriviteraStats2 7.E.025.

A population of scores is normally distributed with a standard deviation equal to 6. State whether the standard error will increase, decrease, or remain unchanged if the value of the population standard deviation is changed to the following.

Part (a)

σ = 8