Deck 7: Analyzing Proportions

Question 1

The probability distribution for the number of "successes" in a fixed number of independent trials, when the probability of success is the same for each, is called which of the following?

A) Binomial distribution
B) Bivariate distribution
C) Joint distribution
D) Pairwise distribution

Answer 1

A

Question 2

Which of the following is not an assumption of the binomial distribution?

A) All trials are independent of the others.
B) The number of trials is a fixed value.
C) The probability of success is less than 0.5.
D) The same probability of success for each trial.

Answer 2

C

Question 3

The value of 5! is which of the following?

A) 20
B) 120
C) 125
D) 225

Answer 3

B

Question 4

The value of 7! is which of the following?

A) 1,040
B) 1,840
C) 5,040
D) 12,840

Question 5

The value of 8! / 5! is which of the following?

A) 40
B) 336.
C) 1,300
D) 14,630

Question 6

The value of 25! / 22! is which of the following?

A) 550
B) 2,300
C) 13,800
D) 215,225

Question 7

The value of 99! / 97! is which of the following?

A) 9,405
B) 9,504
C) 9,603
D) 9,702

Question 8

The value of $\left(\begin{array}{c}11 \\7\end{array}\right)$ is which of the following?

A) 77
B) 330
C) 7,920
D) 37,290

Question 9

The value of $\left(\begin{array}{c}20 \\3\end{array}\right)$ is which of the following?

A) 1,020
B) 1,140
C) 1,260
D) 1,380

Question 10

Consider a class of 10 students in a school district with a 20% prevalence of students with special needs. Assuming the binomial distribution is appropriate, what is the probability that exactly 2 of those students have special needs?

A) 0.201
B) 0.302
C) 0.403
D) 0.504

Question 11

Consider a set of nine separate water samples from a region where the prevalence of bacterial contamination is 30%. Assuming the binomial distribution is appropriate, what is the probability that exactly three of the samples indicate contamination?

A) 0.267
B) 0.333
C) 0.350
D) 0.400

Question 12

Imagine that we are rolling a six-sided die and we do that seven times. What is the probability that we roll a "one" either six or seven times?

A) 0.00010
B) 0.00011
C) 0.00012
D) 0.00013

Question 13

Consider a bird has laid a clutch of six eggs that all hatch. Assuming the binomial distribution is appropriate, what is the probability that there is an equal number of male and female chicks that hatch?

A) 0.279
B) 0.313
C) 0.346
D) 0.500

Question 14

Consider a binomial distribution with a sample size of 10 and a success probability of 0.9. What is the probability that there are more than 8 successes?

A) 0.193
B) 0.349
C) 0.387
D) 0.736

Question 15

What is the standard error of the proportion when the sample size is 55 values and the sample proportion is 0.4?

A) 0.033
B) 0.066
C) 0.100
D) 0.133

Question 16

What is the standard error of the proportion when the sample size is 81 values and the sample proportion is 0.35?

A) 0.023
B) 0.033
C) 0.043
D) 0.053

Question 17

If we have a sample with 25 values and a sample proportion of 0.36, what would the standard error of the proportion be?

A) 0.072
B) 0.096
C) 0.177
D) 0.215

Question 18

If we have a sample with 50 values and a sample proportion of 0.40, what would the standard error of the proportion be?

A) 0.049
B) 0.059
C) 0.069
D) 0.079

Question 19

Consider the claim that 60% of the members of a population of bacteria have a plasmid conferring antibiotic resistance. If we collected 11 bacterial samples and 3 proved to have the resistance plasmid, what would the P-value of a binomial test of this hypothesis be?

A) 0.0233
B) 0.0467
C) 0.1774
D) 0.3547

Question 20

Consider the claim that 60% of the members of a population of bacteria have a plasmid conferring antibiotic resistance. If we collected 11 bacterial samples and 4 proved to have the resistance plasmid, what would the P-value of a binomial test of this hypothesis be?

A) 0.0666
B) 0.0701
C) 0.1401
D) 0.3636

Question 21

Consider the claim that 60% of the members of a population of bacteria have a plasmid conferring antibiotic resistance. If we collected some random bacterial samples and the P-value of a binomial test was 0.015, what would our conclusion be?

A) We fail to reject the null hypothesis and therefore conclude that the true proportion of bacteria with resistance plasmids differs from 60%.
B) We fail to reject the null hypothesis and therefore conclude that the true proportion of bacteria with resistance plasmids does not differ from 60%.
C) We reject the null hypothesis and therefore conclude that the true proportion of bacteria with resistance plasmids differs from 60%.
D) We reject the null hypothesis and therefore conclude that the true proportion of bacteria with resistance plasmids does not differ from 60%.

Question 22

Consider the claim that 60% of the members of a population of bacteria have a plasmid conferring antibiotic resistance. If we collected some random bacterial samples and the P-value of a binomial test was 0.15, what would our conclusion be?

A) We fail to reject the null hypothesis and therefore conclude that the true proportion of bacteria with resistance plasmids differs from 60%.
B) We fail to reject the null hypothesis and therefore conclude that the true proportion of bacteria with resistance plasmids does not differ from 60%.
C) We reject the null hypothesis and therefore conclude that the true proportion of bacteria with resistance plasmids differs from 60%.
D) We reject the null hypothesis and therefore conclude that the true proportion of bacteria with resistance plasmids does not differ from 60%.

Question 23

Imagine a surgery that is known to have a 10% chance of serious side effects. An internal hospital review shows that three out of eight of a particular doctor's patients have these side effects. If we want to know whether this doctor's patients are experiencing unusually low or high rates of side effects, what would the P-value of a binomial test of this hypothesis be?

A) 0.022
B) 0.044
C) 0.066
D) 0.088

Question 24

Imagine a surgery that is known to have a 10% chance of serious side effects. An internal hospital review shows that three out of eight of a particular doctor's patients have these side effects. If we conducted a binomial test of whether this doctor's patients are experiencing unusually low or high rates of side effects, what would our conclusion be?

A) We fail to reject the null hypothesis and therefore conclude that the rate of side effects for this doctor differs from the usual 10%.
B) We fail to reject the null hypothesis and therefore conclude that the rate of side effects for this doctor does not seem to differ from the usual 10%.
C) We reject the null hypothesis and therefore conclude that the rate of side effects for this doctor differs from the usual 10%.
D) We reject the null hypothesis and therefore conclude that the rate of side effects for this doctor does not seem to differ from the usual 10%.

Question 25

Imagine a surgery that is known to have a 10% chance of serious side effects. An internal hospital review shows that four out of eight of a particular doctor's patients have these side effects. If we want to know whether this doctor's patients are experiencing usually low or high rates of side effects, what would the P-value of a binomial test of this hypothesis be?

A) 0.009
B) 0.09
C) 0.9
D) 0.5

Question 26

Imagine a surgery that is known to have a 10% chance of serious side effects. An internal hospital review shows that 4 out of 8 of a particular doctor's patients have these side effects. If we conducted a binomial test of whether this doctor's patients are experiencing usually low or high rates of side-effects, what would our conclusion be?

A) We fail to reject the null hypothesis and therefore conclude that the rate of side effects for this doctor differs from the usual 10%.
B) We fail to reject the null hypothesis and therefore conclude that the rate of side effects for this doctor does not seem to differ from the usual 10%.
C) We reject the null hypothesis and therefore conclude that the rate of side effects for this doctor differs from the usual 10%.
D) We reject the null hypothesis and therefore conclude that the rate of side effects for this doctor does not seem to differ from the usual 10%.

Question 27

Using the Agresti-Coull method, what is the 95% confidence interval for the proportion when there are 15 observed successes and 25 observed failures?

A) 0.243 < p < 0.529
B) 0.253 < p < 0.519
C) 0.263 < p < 0.509
D) 0.273 < p < 0.499

Question 28

Using the Wald method, what is the 95% confidence interval for the proportion when there are 15 observed successes and 25 observed failures?

A) 0.240 < p < 0.540
B) 0.235 < p < 0.535
C) 0.230 < p < 0.530
D) 0.225 < p < 0.525

Question 29

What is the 95% confidence interval, using the Agresti-Coull method, for the proportion when there are 7 observed successes out of a total of 50 trials?

A) 0.014 < p < 0.213
B) 0.044 < p < 0.236
C) 0.067 < p < 0.266
D) 0.104 < p < 0.282

Question 30

What is the 95% confidence interval, using the Wald method, for the proportion when there are 7 observed successes out of a total of 50 trials?

A) 0.014 < p < 0.213
B) 0.044 < p < 0.236
C) 0.074 < p < 0.259
D) 0.104 < p < 0.282

Question 31

If a study reveals five successes and eight failures, in how many different ways (i.e., sequences) could this have occurred??

A) 789
B) 839
C) 1,287
D) 1,716

Question 32

If a study reveals 10 successes and 12 failures, in how many different ways (i.e., sequences) could this have occurred??

A) 646,646
B) 705,432
C) 801,426
D) 953,664

Question 33

If a study with a total sample size of 14 measures 7 successes, in how many different sequences could these successes have occurred??

A) 3,003
B) 3,432
C) 3,867
D) 4,124

Question 34

If a study with a total sample size of 15 measures 6 successes, in how many different sequences could these successes have occurred??

A) 2,002
B) 3,003
C) 4,004
D) 5,005

Question 35

When calculating binomial probabilities, we use the term "success" for the desired outcome of each individual trial.

Question 36

20! > 99²

Question 37

10! < 1,000

Question 38

The term $\left(\begin{array}{c}10 \\4\end{array}\right)$ is stated verbally as "ten choose four."

Question 39

The term $\left(\begin{array}{l}7 \\3\end{array}\right)$ is stated verbally as "three choose seven."

Question 40

Given a fixed proportion of successes in the population, the width of the sampling distribution for the number of successes gets narrower as the sample size gets larger.

Question 41

Given a fixed proportion of successes in the population, the width of the sampling distribution for the proportion of successes gets narrower as the sample size gets larger.

Question 42

The binomial test uses data to test whether a sample proportion matches the null expectation for the proportion.

Question 43

When the binomial test returns a P-value less than 0.05, that generally means that the data match the expectations arising from using the binomial distribution to model the population.

Question 44

If a binomial test returns a P-value greater than 0.05, we typically interpret this as meaning there is a match between the data and our expectations arising from using the binomial distribution to model the population.

Question 45

The standard error of a proportion is used to estimate how much variation there is in the sample data.

Question 46

When calculating the standard error of a proportion, if the sample size is small or the population proportion is close to 0 or 1, then the Wald method is preferable to the Agresti-Coull method.

Question 47

If the 95% confidence interval for the proportion does not include the value hypothesized in the binomial test, then the test will almost certainly return a P-value greater than 0.05.

Question 48

When we do a binomial test and obtain a P-value smaller than 0.05, then the 95% confidence interval for the proportion will almost certainly not include the proportion we hypothesized.

Question 49

Draw a bar chart showing the sampling distribution for the proportion of successes for sample size 3 where the population proportion of successes is 0.6. Clearly label each axis and be precise.

Question 50

Clearly and precisely describe the statistical conclusions that we make when conducting a binomial test. Describe both results: when we obtain a P-value less than 0.05 and when we obtain a P-value larger than 0.05.

Question 51

Consider a situation in which a group of physiologists are studying the sizes of wings in seagulls. They are interested in whether the gulls show asymmetry or symmetry with respect to the wing sizes. They humanely trap (and subsequently release) 14 seagulls, and after measuring the wings they determine that 10 had larger right wings than left wings and 4 had larger left wings than right wings. Use a binomial test and the calculation of 95% confidence intervals to address this question: Is there sufficient evidence to determine whether the population of seagulls is symmetric with respect to their wing sizes or not?

Question 52

Consider a situation in which a group of ecologists are studying the frequency of symbiotic relationships between acacia trees and ant colonies in Bolivia. Previous studies in Colombia have shown the frequency of symbiotic relationships there to be 30%, and the ecologists are interested in whether the same frequency holds in Colombia. They locate 20 trees and determine that 11 of the trees appear to have a symbiotic relationship with an ant colony. Use a binomial test and the calculation of 95% confidence intervals to address this question: Is there sufficient evidence to determine whether the frequency of symbiotic acacia trees in Colombia differs from 30% or not?

Question 53

Demonstrate with a hypothetical numerical example how the Agresti-Coull method generates a narrower confidence interval than the Wald method.

Question 54

What is the relationship between the 95% confidence interval for a proportion and the P-value of a binomial test?