Question

In a batch of 26 pedometers, 3 are believed to be defective. A quality control engineer randomly selects 7 units to test. Let random variable X=the number of defective units that are among the 7 units tested. a. Find the probability mass function f(x)=P(X =x), and sketch its histogram. b. Find P(X = 1). What does this number represent? c. Find P(X 21). What does this number represent? C. Using the hypergeometric probability distribution model, set up an expression that can be used to find a single ordered pair in the probability mass function f(x)=P(X = x). f(x)=P(X =x) = ) PRE (Simplify your answers.) a. Find the probability mass function f(x)=P(X=x). f(x)= O (Type an ordered pair. Use a comma to separate answers as needed. Round to five decimal places as needed.) Let x =the number of defective pedometers among the 7 units tested, and let y=f(x). Choose the correct histogram below. O A. OB. OC. OD. 0.7 0.6 0.5 AY 0. 0.6 0.5 0. 0. 0.2 0. 0 0 1 2 3 TTTTT 0.3 0.2 0.1- o 0.6 0.5 0.4 0.3 0.2 0.1- 0.7 0.6 0.5-1 04- 03 0.2 0 1 0 0 0 1 2 3 b. Find P(X = 1). P(X = 1) = (Round to five decimal places as needed.) What does P(X = 1) represent? O A. This is the probability that at least one of the 7 pedometers chosen for inspection was defective. O B. This is the probability that only 1 out of the 26 pedometers was defective. O C. This is the probability that exactly one of the 7 pedometers chosen for inspection was defective. OD. This is the probability that the first of the 7 pedometers chosen for inspection was defective. c. Find P(X 21). P(X21)=(Round to five decimal places as needed.) What does P(X21) represent? O A. This is the probability that exactly one of the 7 pedometers chosen for inspection was defective. O B. This is the probability that at least 1 out of the 26 pedometers was defective. C. This is the probability that at least one of the 7 pedometers chosen for inspection was defective. OD. This is the probability that some pedometers after the first of the 7 pedometers chosen for inspection were defective.

Answer 1

a. The probability mass function (pmf) f(x) of a random variable X is calculated using the hypergeometric distribution formula. b. P(X = 1) represents the probability of selecting exactly 1 defective pedometer out of the 7 tested. c. P(X > 2) represents the probability of selecting more than 2 defective pedometers out of the 7 tested.

Step-by-step explanation:

a. Probability Mass Function

The probability mass function (pmf) f(x) of a random variable X is defined as the probability that X takes on a specific value x. In this case, X represents the number of defective pedometers among the 7 units tested.

To find the pmf, we can use the hypergeometric distribution formula:

f(x) = (C(D, x) * C(N-D, n-x)) / C(N, n)

Where:

• D = number of defective pedometers (3)
• N = total number of pedometers (26)
• n = number of pedometers tested (7)
• C(a, b) = combination formula (a choose b)

We can calculate the pmf for all values of x from 0 to 7.

b. P(X = 1)

To find P(X = 1), we substitute x = 1 into the pmf formula:

f(1) = (C(3, 1) * C(26-3, 7-1)) / C(26, 7)

This represents the probability of selecting exactly 1 defective pedometer out of the 7 tested.

c. P(X > 2)

To find P(X > 2), we need to sum up the probabilities of X being greater than 2:

P(X > 2) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)

This represents the probability of selecting more than 2 defective pedometers out of the 7 tested.