Question 1

A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts. Prices for 100 rats follow the following distribution: 
$\begin{array}{lrrr}
\text { Price: }& \$ 10.00 & \$ 12.50 & \$ 15.00 \
\hline \text { Probability: }&0.35 & 0.40 & 0.25
\end{array}$
How much should the lab budget for next year's rat orders be, assuming this distribution does not change?
A) $520
B) $637
C) $650
D) $780

Accepted Answer

The lab orders 100 rats a week, so the total number of rats ordered in a year is:
$100	ext{ rats/week}	imes52	ext{ weeks}=5200	ext{ rats}$
To find the expected cost of the rats, we use the probabilities given in the table:
Expected cost = (\$10.00 $	imes$ 0.35) + (\$12.50 $	imes$ 0.40) + (\$15.00 $	imes$ 0.25) = \$11.75
So the total budget for next year would be:
$5200	ext{ rats}	imes\$11.75/	ext{100 rats}= \$611$
However, we need to round up to the nearest dollar, so the answer is:
Answer: B) $637

Question 2

Suppose that a judge's decisions follow a binomial distribution and that his verdict is incorrect 10% of the time. In his next 10 decisions, the probability that he makes fewer than 2 incorrect verdicts is 0.736.

Accepted Answer

This can be solved using the binomial distribution formula, but it is easier to use a cumulative distribution function (CDF) calculator. Using a binomial CDF calculator with n = 10, p = 0.1, and x = 1, we get a probability of 0.264. Subtracting this from 1 gives us 0.736, which matches the given probability. Therefore, the statement is true.

Question 3

The largest value that a Poisson random variable X can have is n.

Accepted Answer

A Poisson random variable can theoretically take any non-negative integer value, though values beyond a certain point become increasingly unlikely. So there is no maximum value for a Poisson random variable.

Question 4

In a Poisson distribution, the mean and standard deviation are equal.

Accepted Answer

The answer of In a Poisson distribution, the mean and...

Question 5

A multiple-choice test has 30 questions. There are 4 choices for each question. A student who has not studied for the test decides to answer all questions randomly. What type of probability distribution can be used to figure out his chance of getting at least 20 questions right?
A) Binomial distribution
B) Poisson distribution
C) Hypergeometric distribution
D) None of the above.

Accepted Answer

The binomial distribution is appropriate for this situation because it involves a fixed number of independent trials (answering each question) with a binary outcome (correct or incorrect) and a constant probability of success (1/4 chance of selecting the correct answer). Additionally, we are interested in the number of successes in a fixed number of trials (getting at least 20 questions right out of 30), which fits the criteria of a binomial distribution.

Question 6

In a Poisson distribution, the mean and variance are equal.

Accepted Answer

This is true. It is one of the defining characteristics of a Poisson distribution.

Question 7

If π remains constant in a binomial distribution, an increase in n will not change the mean.

Accepted Answer

The answer of If π remains constant in a binomial...

Question 8

If π remains constant in a binomial distribution, an increase in n will increase the variance.

Accepted Answer

The variance of a binomial distribution is given by np(1-p), where n is the sample size and p is the probability of success. If π remains constant (i.e., p remains constant), an increase in n will increase the variance. This is because as the sample size increases, there is more variability in the data and therefore a larger spread of possible outcomes.

Question 9

The number of males selected in a sample of 5 students taken without replacement from a class of 9 females and 18 males has a binomial distribution.

Accepted Answer

The number of males selected in a sample of 5 students taken without replacement would have a hypergeometric distribution, not a binomial distribution.

Question 10

The number of customers arriving at a department store in a 5-minute period has a binomial distribution.

Accepted Answer

A binomial distribution requires a fixed number of independent trials with only two possible outcomes (success or failure) and a constant probability of success for each trial. The number of customers arriving in a 5-minute period is not a fixed number of trials and the probability of a customer arriving may change from one 5-minute period to another. Therefore, the number of customers arriving in a 5-minute period does not have a binomial distribution.

Question 11

A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts. Suppose the mean cost of rats used in lab experiments turned out to be $13.00 per week. Interpret this value.
A) Most of the weeks resulted in rat costs of $13.00.
B) The median cost for the distribution of rat costs is $13.00.
C) The expected or average cost for all weekly rat purchases is $13.00.
D) The rat cost that occurs more often than any other is $13.00.

Accepted Answer

The mean cost of rats used in lab experiments is the average cost for all weekly rat purchases.

Question 12

The local police department must write, on average, 5 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.5 tickets per day. Interpret the value of the mean.
A) The number of tickets that is written most often is 6.5 tickets per day.
B) Half of the days have less than 6.5 tickets written and half of the days have more than 6.5 tickets written.
C) If we sampled all days, the arithmetic average or expected number of tickets written would be 6.5 tickets per day.
D) The mean has no interpretation since 0.5 ticket can never be written.

Accepted Answer

The mean of a Poisson distribution represents the expected or average number of events (in this case, tickets written) in a given time period. Therefore, option C is correct stating that if we sampled all days, the arithmetic average or expected number of tickets written would be 6.5 tickets per day. Option A is incorrect because the mean is not necessarily the value that is written most often. Option B is also incorrect because the distribution is not necessarily symmetric, and the mean does not necessarily divide the distribution in half. Option D is incorrect because the fact that 0.5 ticket cannot be written does not invalidate the interpretation of the mean.

Question 13

The diameters of 10 randomly selected bolts have a binomial distribution.

Accepted Answer

The diameters of bolts cannot follow a binomial distribution, as it is not a discrete variable with a fixed number of trials. It is a continuous variable that can take on any value within a certain range.

Question 14

A financial analyst is presented with information on the past records of 60 start-up companies and told that in fact only 3 of them have managed to become highly successful. He selected 3 companies from this group as the candidates for success. To analyze his ability to spot the companies that will eventually become highly successful, he will use what type of probability distribution?
A) Binomial distribution
B) Poisson distribution
C) Hypergeometric distribution
D) None of the above.

Accepted Answer

The situation described involves a finite population (60 start-up companies) from which a sample (3 companies) is selected, without replacement, to determine the proportion of highly successful companies in the entire population. This is best represented by a hypergeometric distribution. The binomial distribution would be used if the sample size was fixed (e.g. analyzing the success rate of a single company over a period of time with a fixed number of trials) and the Poisson distribution would be used if the events being analyzed occurred randomly over time and space (e.g. analyzing the number of customer complaints received per day by a restaurant).

Question 15

The Poisson distribution can be used to model a continuous random variable.

Accepted Answer

The Poisson distribution is used to model discrete random variables, not continuous random variables.

Question 16

Another name for the mean of a probability distribution is its expected value.

Accepted Answer

This is correct, the mean and expected value of a probability distribution are interchangeable terms.

Question 17

On the average, 1.8 customers per minute arrive at any one of the checkout counters of a grocery store. What type of probability distribution can be used to find out the probability that there will be no customer arriving at a checkout counter?
A) Binomial distribution
B) Poisson distribution
C) Hypergeometric distribution
D) None of the above.

Accepted Answer

The Poisson distribution is commonly used for modeling the number of arrivals in a fixed time interval, given a known average rate of arrivals. In this case, we know that the average rate is 1.8 customers per minute, and we want to find the probability of zero arrivals. Therefore, the Poisson distribution is the appropriate choice. The other options (binomial and hypergeometric distributions) are not appropriate because they are used for discrete outcomes with different assumptions about the population and sampling process.

Question 18

The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.

Accepted Answer

The Poisson distribution models the number of occurrences of an event in a fixed interval of time, given that the events occur independently of each other and with a known average rate. Therefore, it is appropriate for modeling the arrival of customers in a department store in a fixed time period.

Question 19

Suppose that the number of airplanes arriving at an airport per minute is a Poisson process. The average number of airplanes arriving per minute is 3. The probability that exactly 6 planes arrive in the next minute is 0.05041.

Accepted Answer

The answer of Suppose that the number of airplanes arriving...

Question 20

The number of males selected in a sample of 5 students taken without replacement from a class of 9 females and 18 males has a hypergeometric distribution.

Accepted Answer

A hypergeometric distribution is used in situations where sampling is done without replacement from a finite population of size N containing a subpopulation of size K. In this case, the sample of 5 students is taken without replacement from a population of 27 students (9 females and 18 males), and the number of males selected in the sample is of interest. Therefore, the distribution of the number of males selected in the sample is hypergeometric.

Quiz 5: Discrete Probability Distributions