Solved on Feb 08, 2024

Find P(X=19)P(X=19) for the given sampling distribution: X={16,12,7,10,19}X = \{-16, -12, -7, 10, 19\} and P(X)={1/100,1/50,9/100,3/50,?}P(X)= \{1/100, 1/50, 9/100, 3/50, \text{?}\}.

STEP 1

Assumptions
1. The values of XX represent the possible outcomes of a random variable.
2. P(X)P(X) represents the probability of each outcome.
3. The probabilities of all possible outcomes must sum up to 1 because the total probability of all outcomes in a probability distribution is always 1.

STEP 2

Calculate the sum of the given probabilities.
P(X)=1100+150+9100+350\sum P(X) = \frac{1}{100} + \frac{1}{50} + \frac{9}{100} + \frac{3}{50}

STEP 3

Find a common denominator to add the fractions. The common denominator for 100 and 50 is 100.
1100=1100\frac{1}{100} = \frac{1}{100} 150=2100\frac{1}{50} = \frac{2}{100} 9100=9100\frac{9}{100} = \frac{9}{100} 350=6100\frac{3}{50} = \frac{6}{100}

STEP 4

Add the fractions with the common denominator.
P(X)=1100+2100+9100+6100\sum P(X) = \frac{1}{100} + \frac{2}{100} + \frac{9}{100} + \frac{6}{100}

STEP 5

Perform the addition.
P(X)=1+2+9+6100\sum P(X) = \frac{1 + 2 + 9 + 6}{100} P(X)=18100\sum P(X) = \frac{18}{100}

STEP 6

Simplify the fraction.
P(X)=18100=950\sum P(X) = \frac{18}{100} = \frac{9}{50}

STEP 7

Subtract the sum of the given probabilities from 1 to find the missing probability P(X=19)P(X=19).
P(X=19)=1P(X)P(X=19) = 1 - \sum P(X)

STEP 8

Plug in the calculated sum of probabilities into the equation.
P(X=19)=1950P(X=19) = 1 - \frac{9}{50}

STEP 9

Find a common denominator to subtract the fractions. The common denominator for 1 and 50 is 50.
1=50501 = \frac{50}{50}

STEP 10

Perform the subtraction.
P(X=19)=5050950P(X=19) = \frac{50}{50} - \frac{9}{50}

STEP 11

Calculate the result.
P(X=19)=50950P(X=19) = \frac{50 - 9}{50} P(X=19)=4150P(X=19) = \frac{41}{50}

STEP 12

Convert the fraction to a decimal to match the answer choices.
P(X=19)=4150=0.82P(X=19) = \frac{41}{50} = 0.82
The probability P(X=19)P(X=19) is 0.82.

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