stats

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Math 1350 (Stat 251), Section 10, fall 2021, Exam # 2, October 6, 2021 If you use the calculator to get your answer, write the name of the function and the values of the arguments used, e. g., Normalcdf (95, 105, 100, 15). If the problem has units, don’t forget to state them.]

Name ____________________________________

1. A boeing 747 has 3 engines. Assume the probability of any one engine failing on a 500-mile flight is 0.0005 = 5.0 * 10-4. Assuming statistical independence, what is the probability of all 3 engines failing on a 500-mile flight? [7 points]

------------------------------------------------------------------------------------------------------------------------------- ---- 2. The odds against horse Swayback winning a particular race are given as 4 to 15 against.

Using those odds, what is the probability that Swayback will win the race? [6 points]

------------------------------------------------------------------------------------------------------------------------------- -- 3. What is meant by subjective probability? When might it be used to assign a probability? [6 points]

----------------------------------------------------- ---------------------------------------------------------- 4. Give an example of a) a simple event, and [ 3 points each]

b) a compound event.

------------------------------------------------------------------------------------------------------------------------------- -------- 5. A study conducted several years ago, called the Hanes study, compared the heights of men and of women. According to the study, the mean value of women’s heights in the study was 63.5 inches, and the standard deviation was 2.5 inches. For men, the mean value of heights was 69 inches, and the standard deviation was 3 inches. Suppose a woman is 68 inches tall and her husband is 74 inches tall. Use the z-statistic to determine which of the two is relatively taller among his or her group, the woman or the man. [7 points]

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------------------------------------------------------------------------------------------------------------------------------- ---- 6. Suppose a basket contains 10 red balls and 4 green balls. One ball is drawn (without looking into the basket), its color noted. Then another is drawn without replacing the first ball. What is the probability

a) of the compound event of drawing a red ball first and then a green ball? Write your answer as a fraction in lowest terms. [2 points each]

b) of drawing a green ball first and then a red ball? Write your answer as a fraction in lowest terms.

c) of drawing one red ball and one green ball in any order? That is, the probability either first red and 2nd green, OR first green and 2nd red.

d) of drawing two balls of the same color? This is the complement of the event in c) Show your calculations on all four questions! To calculate d) directly, the probability of drawing 1st red and 2nd red, or drawing 1st green and 2nd green.

------------------------------------------------------------------------------------------------------------------------------- ---- 7. Calculate the arithmetic mean, median, and midrange of the following numbers: 17, 26, 36, 50, 54, 73, and 87. [3, 2, and 3 points]

------------------------------------------------------------------------------------------------------------------------------- ---- 8. Suppose Army recruits have a mean IQ of 100 and a standard deviation of 10.

a) Using the empirical rule, what is the approximate percentage of recruits with IQs of between 80 and 120? [4 points each]

b) Using the range rule of thumb, is an IQ of 78 unusual?

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------------------------------------------------------------------------------------------------------------------------------- ---- 9. Group [5 points each]

O A B AB Total Type Rh+ 47 35 12 6 100 Rh- 7 6 3 2 18 54 41 15 8 118

a) If one person is randomly selected, find the probability of getting someone who is group B and type Rh-.

b) If one person is randomly selected, find the probability of getting someone who is group B or type Rh-.

10. Using the table in # 9, what is the probability that [5 points each] a) The person is Rh-, given that he/she is of type A?

b) The person is of type A, given that he/she is of Rh-?

------------------------------------------------------------------------------------------------------------------------------- ---- 11. Assume that the probability of event A occurring on a single trial is 0.8. Denote the complement of A by Ac. What is the probability of event Ac occurring at least once in 7 independent trials? The complement of this event is 7 A’s in 7 independent trials, or no Ac’s in 7 independent trials. [7 points]

------------------------------------------------------------------------------------------------------------------------------- ---- 12. The current Senate of the United States includes 76 male Senators and 24 female Senators. a) If one of these Senators is randomly selected, what is the probability that a woman is selected? [5 points]

b) Does this support the claim that men and women have the same chance of being elected as Senators? [3 points]

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------------------------------------------------------------------------------------------------------------------------------- ---- 13 a) – e). True or false? P means probability. The other capital letters stand for events. P (R | S) is read “the conditional probability that R occurs given that S has occurred.” [2 points each]

a) P (R | S) = P (R and S) = P (R ∩ S) P (S) P (S)

b) P (R and S) = P (R | S) * P (R)

c) P (D and E) = P (D ∩ E) = P (D) * P (E) if D and E are statistically independent.

d) P (R | S) = P (S | R)

e) P (A or B) = P (A) + P (B) - P (A and B)