Q: A company classifies its customers into four age groups with the following probabilities of reporting at least one accident in a year: $$0.09, 0.14, 0.16, 0.08. $$The proportions of customers in each group are $$0.15, 0.34, 0.36, 0.15$$ respectively. What is the probability that a randomly selected customer reports at least one accident in a year?

$$P(A) = 0.09 \times 0.15 + 0.14 \times 0.34 + 0.16 \times 0.36 + 0.08 \times 0.15 = 0.0607$$

Q: Given the data set $$2, 6, 0, 1, 0, 2$$, what is the coefficient of variation (CV)?

$$CV = \frac{2.2286}{1.16} \times 100 = 121.56$$

Q: A customer is randomly selected from an insurance company. What is the probability that the customer is from age group 1, given that they reported at least one accident during the year?

$$P(C_1$ | A) = \frac{0.09 \times 0.15}{0.0607} = 0.2224$$

Question 1

A machine produces pieces whose lengths are normally distributed with mean $$\mu = 9 cm $$and variance $$\$$sigma^2$$ = 12.25 cm$^2. $$What is the probability that a randomly selected piece is longer than 11 cm?

Accepted Answer

$$P(X \u003e 11) = 1 - P\left(Z \u003c \frac{11-9}{3.5}\right) = 1 - P(Z \u003c 0.57) = 0.2843$$

Question 2

Given the data set: $$2, 6, 0, 1, 0, 2$$, what is the median?

Accepted Answer

The median is $$1.5$$

Question 3

A company classifies its customers into four age groups with the following probabilities of reporting at least one accident in a year: $$0.09, 0.14, 0.16, 0.08. $$The proportions of customers in each group are $$0.15, 0.34, 0.36, 0.15$$ respectively. What is the probability that a randomly selected customer reports at least one accident in a year?

Accepted Answer

$$P(A) = 0.09 	imes 0.15 + 0.14 	imes 0.34 + 0.16 	imes 0.36 + 0.08 	imes 0.15 = 0.0607$$

Question 4

For a binomial random variable $$X \sim 	ext{Bin}(2, 2/3)$$, what is the probability that at least one engine functions throughout the flight?

Accepted Answer

$$P(X \geq 1) = 1 - (1/3)^2$ = 8/9$$

Question 5

Given a normal distribution $$X \sim N(9, 12.25)$$, what is the probability that a piece is shorter than 7.5 cm?

Accepted Answer

$$P(X \u003c 7.5) = P\left(Z \u003c \frac{7.5-9}{3.5}\right) = P(Z \u003c -0.43) = 0.3336$$

Question 6

A quality controller asserts that a piece is considered unacceptable if its length exceeds 12 cm or is less than 5.5 cm. What fraction of the produced pieces will be considered as good quality pieces?

Accepted Answer

$$P(5.5 \u003c X \u003c 12) = P(X \u003c 12) - P(X \u003c 5.5) = 0.8577 - 0.0718 = 0.7859$$

Question 7

Given the data set $$2, 6, 0, 1, 0, 2$$, what is the coefficient of variation (CV)?

Accepted Answer

$$CV = \frac{2.2286}{1.16} 	imes 100 = 121.56$$

Question 8

Suppose $$X is $$the largest of two numbers observed on randomly drawn balls, each numbered 1 to 6. What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 3.5$$

Question 9

For the random variable $$X in $$the previous question, what is the probability that $$X$$ takes a value in the interval $$[E(X) - \sigma(X), E(X) + \sigma(X)]$$?

Accepted Answer

$$P(X = 2) + P(X = 3) = 0.5$$

Question 10

A company wants to ensure that only 1.7% of produced pieces are longer than 14 cm. If the lengths are normally distributed with mean $$9 cm$$, what should the standard deviation $$\sigma be$$?

Accepted Answer

$$\sigma = 2.3588$$

Question 11

If the probability that an aircraft engine fails is $$1/3$$ and a 4-engine aircraft completes its flight if at least 2 engines function, what is the probability of a safe flight?

Accepted Answer

$$P(Y \geq 2) = C_4$^2 (2/3)^2$ (1/3)^2$ + C_4$^3 (2/3)^3$ (1/3) + (2/3)^4$ = 0.8395$$

Question 12

Given a random variable $$X$$ with $$E(X) = 3.5$$ and $$Var(X) = 1.89297$$, what is $$\sigma(X)$$?

Accepted Answer

Question 13

A customer is randomly selected from an insurance company. What is the probability that the customer is from age group 1, given that they reported at least one accident during the year?

Accepted Answer

$$P(C_1$ | A) = \frac{0.09 	imes 0.15}{0.0607} = 0.2224$$

Question 14

What is the 25th percentile of the data set $$2, 6, 0, 1, 0, 2$$?

Accepted Answer

Question 15

If $$X \sim N(9, 12.25)$$, what is $$P(5.5 \u003c X \u003c 12)$$?

Accepted Answer

$$P(5.5 \u003c X \u003c 12) = 0.7859$$

Question 16

A customer is randomly selected from an insurance company. What is the probability that the customer is older than 65 and has not reported an accident during the year?

Accepted Answer

Question 17

Given a normal distribution $$X \sim N(9, 12.25)$$, what is the probability that a piece is longer than 14 cm?

Accepted Answer

$$P(X \u003e 14) = 0.017$$

Question 18

If the probability that a customer under 45 reports at least one accident in a year is $$0.0664$$, what is the probability that a customer over 45 reports at least one accident in a year?

Accepted Answer

$$P(A | C_3$ \cup C_4$) = 0.1567$$

Probability and Statistics: Key Concepts, Applications, and Problem Solutions

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