Probability Questions
1.
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.)
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
2.
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)^{2}p | |
μ =
σ^{2} |
(b) the variance of a probability distribution
σ^{2} =
(x − μ)^{2}p | |
σ^{2} =
xp | |
σ^{2} =
x − p n − 1 |
σ^{2} =
σ |
(c) the standard deviation of a probability distribution
σ =
(x − μ)^{2}p | |
σ =
x − p n − 1 |
σ =
σ^{2} |
σ =
xp | |
3.
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?
4.
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
5.
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?
6.
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) =
7.
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?
8.
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
9.
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
The standard error will increase. The standard error will decrease. The standard error will remain unchanged.
- Part (b)
σ = 2
The standard error will increase. The standard error will decrease. The standard error will remain unchanged.
- Part (c)
σ =
24 |
4 |
The standard error will increase. The standard error will decrease. The standard error will remain unchanged.
- Part (d)
σ =
4 |
16 |
The standard error will increase. The standard error will decrease. The standard error will remain unchanged.
- Part (e)
σ = 6.5
The standard error will increase. The standard error will decrease. The standard error will remain unchanged.
- Part (f)
σ = 0
The standard error will increase. The standard error will decrease. The standard error will remain unchanged.