 # Statistics and Probability Problems

## Question

Part 1
1) --- A variable that has a single numerical value, determined by chance, for each outcome of a procedure is a:
A) Continuous variable B) Random variable C) Unusual variable D) Expected variable
2) ___ All of the following are requirements for a Probability Distribution except:
A) The sum of all probabilities must be 1.
B) A probability can be zero.
C) A probability can be negative.
D) A probability can be 1.

3) __ A probability distribution that meets these requirements: 1) the procedure has a fixed number of trials; 2) the trial must be independent; 3) each trial must have all outcomes classified into two categories; 4) the probability of a success remains the same in all trials is said to be:
A) Independent B) Successful C) Discrete
4) __ All of the following are requirements for the Poisson Distribution except:
A) The occurrences must be random.
B) Success and failure must be equal.
D) Binomial
C) The occurrences must be uniformly distributed over the interval being used.
D) The random variable x is the number of occurrences of an event over some interval.
5) __ If the values are spread evenly over the range of possibilities, a continuous random variable is said to have_ distribution:
A) Uniform B) Dense C) Normal D) Corresponding
6) __ Because the total area under the density curve is equal to 1, there is a correspondence between area and
A) Normality B) Infinity C) Alpha D) Probability
7) __ In a normal distribution curve, if a value is above the mean, we would expect its corresponding z-score to be
A) Positive B) Negative C) Bimodal D) Unusual
8) __ All of the following statistics are said to be unbiased estimators except:
A) Mean B) Range C) Proportion D) Variance
9) __ In a nonstandard normal distribution, the mean is represented with the Greek letter 'mu.' In standard normal distribution, the mean is:
A) the variable x B) a negative z-score C) 99. 7% D) zero
10) __ The data values under a probability curve is said to be:
A) Unusual B) Independent C) Discrete D) Continuous

Part 2
1) In the accompanying table, the random variable x represents the number of books college students read over a semester. Use the probability distribution and your calculator to answer the following questions.

x I P(x)
0    .02
1    .15
2    .29
3    .26
.16
5    .12

a) What is the sum of the probabilities? _____ _
b) What is the mean? ___ _
c) What is the standard deviation? ___ _
d) What is the probability that a student read two books in the semester? __ _
e) What is the probability that a student read four or more books? __ _
f) What is the probability that a student read less than three books? __ _
g) Using the Range Rule of Thumb, what is the maximum usual value? ___ _
h) Using the Range Rule if Thumb, what is the minimum usual value? ___ _
2) Multiple-choice questions on the SAT test each have 5 possible answers (a, b, c, d, e), one of which is correct.
Assume that you guess the answers to 3 such questions.
a) Use the multiplication rule to find the probability that the first guess is correct and the second and third guesses are wrong. That is, find P(CWW) where C denotes a correct answer and W denotes a wrong answer.
b) Beginning with CWW, make a list of the three different possible arrangements of 1 correct answer and 2 wrong answers, then find the probability for each entry in the list.
First possible arrangement:             Probability: __ _
Second possible arrangement: __ _ Probability: __ _
Third possible arrangement: __ _    Probability: __ _
c) Based on the preceding results (part b), what is the probability of getting exactly 1 correct answer when three guesses are made? ___ _

3) The television show NBC Sunday Night Football broadcast a game between the Ravens and the Steelers and received a share of 22, meaning that among the TV sets in use, 22% were tuned to that game (based on Nielsen Media Research). An advertiser wants to obtain a second opinion by conducting its own survey, and a pilot survey begins with 20 households having TV sets in use at the time of that same NBC Sunday Night Football broadcast.
a) The fixed number of trials, n, is: __ _
b) The probability of success, p, is __ _
c) Find the probability that none of the households are tuned to NBC Sunday Night Football. __ _
d) Find the probability that at least one household is tuned in to NBC Sunday Night Football. (Hint: consider the 'compliment' in part c) __ _
e) Find the probability that at most one household is tuned to NBC Sunday Night Football. __ _

4) Mars, Inc. claims that 24% of its M&Ms plain candies are blue. A sample of 100 M&Ms is randomly selected.
a) Find the mean for the numbers of blue M&Ms in such groups of 100. __ _
b) Find the standard deviation for the numbers of blue M&Ms in such groups of 100. __ _
c) We found 27 blue M&Ms in a sample of 100. Is this result unusual (yes or no)? ___ _
5) For Standard Normal Distribution, find the following probabilities based on the area under the bell curve based on the given z-score. You do not need to show a sketch of your curve as your answer, but it may be helpful as part of your own process.
a) Find the area to the left of z-score 0.75:
b) Find the area to the right of z-score 1.24: __ _
c) Find the area to the right of z-score -1.57: ___ _
d) Find the area between z-scores -0.6 and 1.2:
e) Find the area between z-scores -2.87 and 1.34: __ _
f) What is the total area under the curve? __ _

6) For Standard Normal Distribution, find the z-score based on the given area under the bell-curve. You do not need to show a sketch of your curve as your answer, but it is highly-recommended as part of your own process.
a) Find the z-score when the area to the left is 0.9798: __ _
b) Find the z-score when the area to the left is 0.2546: ___ _
c) Find the z-score when the area to the right is 0.1075: __ _
d) Find the z-score when the area to the right is 0. 9418: __ _

7) Adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15. (To complete the following questions, you are encouraged to sketch the bell curve and mark your known values as part of your process. It is not required for you answer.) Be prepared to make a distinction between using z-scores and x-values as needed.
a) Find the probability that a randomly selected adult has an IQ that is less than 115.
b} Find the probability that a randomly selected adult has an IQ greater than 131.5. ___ _
c) Find the probability that a randomly selected adult has an IQ between 110 and 120. ___ _
d) Find P30, which is the IQ score separating the bottom 30% from the top 70%. __ _

8) The ages (years) of the four US Presidents when they were assassinated in office are 56 (Lincoln), 49 (Garfield}, 58 (McKinley}, and 46 (Kennedy). Complete the tables provided and answer the following questions.
a) What is the population mean of the four ages given? __ _
b) Assuming that 2 of the ages are randomly selected with replacement, list the 16 possible samples, their means and their equally-likely probabilities (in your choice of fraction or decimal format).
Sample (n = 2)      Sample Mean    Probability

c) Summarize the sampling distribution of the means and their probabilities from the table above. You will have two lists for input purposes.
Sample Mean       Probability

d) Based on your second table (part c), what is the mean of the sample mean? __ _
e) Does the mean of the sample means target the population mean (yes or no)? ___ _
9) SAT scores are normally distributed with meanµ= 1518 and standard deviation a= 325.
a) If 1 SAT score is randomly selected, find the probability that it is between 1440 and 1480.
P(1440 < x < 1480) ___ _
b) If sample of 16 SAT scores are randomly selected, find the probability that they have a sample mean between 1440 and 1480.
P(1440 < x < 1480) ___ _
c) Short answer (1- 2 sentences): Why can the Central Limit Theorem be used in part (b), even though the sample size does not exceed 30?

10) Apply the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. For example, the probability of "more than 20 defective items" corresponds to the area of the normal curve described with this answer: "the area to the right of 20.5." Your answer should indicate the "area to the right," "area to the left," or "area in between" your continuity correction value(s).
a) Probability of more than 8 Senators who are women:
b) Probability of fewer than 5 passengers who did not show up for a flight:
c) Probability of no more than 15 peas with green pods:
d) Probability that the number of job applicants late for interviews is between 5 and 9 inclusive

## Solution Preview

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Part 1
Question 1: The answer is option B.
Question 2: The answer is option C.
Question 3: The answer is option D.
Question 4: The answer is option B....
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