1. The probability of event A is P(A) = 0.42 and the probability of event B is P(B) = 0.54.

Express all answers in decimal form rounded to **2 decimal places**.

a) Find P(A and B) if A and B are disjoint events.

b) Find P(A or B) if A and B are disjoint events.

c) Find P(A or B) if P(A and B) = 0.15.

d) Find P(A and B) if A and B are independent.

e) Find P(AC), the probability of the complement of event A.

2. The percentage of people who are left handed is 9.2%. Let the variable X represent the number of people that are left-handed in a random sample of 50 people.

a) Find the mean of X. (decimal answer, rounded 1 decimal place)

b) Find the standard deviation of X. (decimal answer, rounded 3 decimal places)

c) Find the probability that no one in the sample is left-handed. (decimal answer, rounded 3 decimal places)

d) Find the probability that at least 1 person in the sample is left-handed. (decimal answer, rounded 3 decimal places)

e) Determine the shape of the distribution for X.

A) skewed-left

B) skewed-right

C) bell-shaped

3.The heights of adult females in the US are normally distributed with mean 65 inches and standard deviation 2.5 inches. Suppose the random variable X represents the height of a female in the US.

a) Find the proportion of US adult females that are between 60 and 66 inches tall. (decimal answer, rounded **3 decimal places**)

b) Find the probability that a randomly selected US adult female is at least 72 inches tall. (decimal answer, rounded **3 decimal places**)

c) Find the proportion of of US adult females that are below 60 inches tall. (decimal answer, rounded **3 decimal places**)

d) Find the 75th percentile for height of US adult females. (decimal answer, rounded **1 decimal place**)

e) Find the height of a US adult female when only 1% of other US adult females are taller. (decimal answer, rounded **1 decimal place**)

4.The following table gives the probability distribution for a random variable X.

x | P(x) |

2 | 0.008 |

3 | 0.076 |

4 | 0.264 |

5 | 0.412 |

6 | 0.240 |

a) Find the mean of X. (decimal answer, rounded **1 decimal place**)

b) Find the standard deviation of X. (decimal answer, rounded **3 decimal places**)

c) Find the probability that X is 2 or 3. (decimal answer, rounded **3 decimal places**)

d) Find the probability that X is at least 4.(decimal answer, rounded **3 decimal places**)

e) Identify the shape of the distribution for X. (letter only, choose one)

A) skewed-left

B) skewed-right

C) bell-shaped

5.The percentage of students with a GPA of 3.0 or higher is 15%. Suppose the random variable X represents the total number of students with a GPA of 3.0 or higher in a random sample of 500 students.

a) Find the mean of X. (round to the nearest **whole number**)

b) Find the standard deviation of X. (round to the nearest **whole number**)

c) Determine the shape of the distribution for X. (letter only)

A) skewed-left, since .5″>

$p$>

.5

and

$n$is small.

B) skewed-right, since

$p$<

.5

and

$n$is small.

C) bell-shaped, since

$p$=

.5

and

$n$is small.

D) normal, since

$n$â‹…

p

â‹…

(

1

âˆ’

p

)

â‰¥

10

.

d) Based on your answers from parts (a)-(c), would it be unusual for 80 students in the sample to have a GPA of 3.0 or higher. (letter only)

A) Yes, since

$P$(

80

)

<

.05

B) Yes, since 80 is more than 2 standard deviations away from the mean

C) No, since 80 is within 2 standard deviations of the mean

D) No, since anyone can get a 3.0 or higher.

E) No, since

$P$(

80

)

â‰¥

.05

.

e) Based on your answers from parts (a)-(c), would it be unusual for 95 students in the sample to have a GPA of 3.0 or higher. (letter only)

A) Yes, since

$P$(

95

)

<

0.05

.

B) Yes, since 95 is more than 2 standard deviations away from the mean.

C) No, since 95 is within 2 standard deviations of the mean.

D) No, since anyone can get a 3.0 or higher.

E) No, since

$P$(

95

)

â‰¥

.05

.

6.An insurance company sells 1-year life insurance policies for 40 year old males. The insurer pays out $10,000 in the event that the insured dies within 1 year of purchasing a policy, and charges a single premium of $30. The probability that a 40 year old male dies within 1 year is 0.23%.

Complete the following probability distribution for X, where X is the amount of money that the insurance company makes on the policy in the events that the man survives the year or dies within the year.

For column x, answers should be integers (positives or negatives of whole numbers; no decimals). For column P(x), answers should be decimals rounded to 4 decimal places. (do not include units for any answers)

event | x | P(x) |

survives | ||

dies |

Find the insurer’s expected value of a 1-year life insurance policy for a 40 year old male. (whole number; round to the **nearest dollar**)

7.Assume that Z is a standard normal random variable.

a) State the mean for Z.

b) State the standard deviation for Z.

c) Find

${}_{}$z

Î±

where

$\mathrm{\xce\pm}$=

0.15

. (decimal answer, round **3 decimal places**)

8.Two fair 6-sided dice are rolled, and the random variable X represents the sum of the values. Complete the following table for the outcomes of X.

The top-most row lists the outcomes for the first die, and the left-most column represents the outcomes for the second die.

Express all answers as **whole numbers**.

1 | 2 | 3 | 4 | 5 | 6 | |

1 | ||||||

2 | ||||||

3 | ||||||

4 | ||||||

5 | ||||||

6 |

9. Answer the following, expressing all answers as decimals, rounded to **3 decimal places**.

Hint: Your answers from question 1 may be useful.

a) For a single fair 6-sided die, find the probability of rolling 6.

b) For a pair of 6-sided dice, find the probability of rolling 6 (the sum of the two dice).

c) For a pair of 6-sided dice, find the probability of rolling at least 3 (the sum of the two dice).