Question

Discrete math The average human lifespan is 79 years with a standard deviation of 15 years. What is the probability that someone will live between 49 and 64 years?

a) 0.1587
b) 0.3413
c) 0.4772
d) 0.6826

Answer 1

The probability that someone will live between 49 and 64 years is approximately 0.4772, corresponding to option c).

Step-by-step explanation:

To find the probability that someone will live between 49 and 64 years, we can use the Z-score formula and the standard normal distribution table. The Z-score is calculated as:

`\[ Z = (X - \mu)/(\sigma) \]`

Where:

-( X ) is the value (in years),

(mu) is the mean (average lifespan), and

(sigma) is the standard deviation.

In this case, (mu = 79) years, ( sigma = 15 ) years, and we want to find the probability for ( X ) between 49 and 64 years.

For ( X = 49 ):

`\[ Z_(49) = (49 - 79)/(15) = -2 \]`

For ( X = 64 ):

`\[ Z_(64) = (64 - 79)/(15) = -1 \]`

Now, we look up the corresponding probabilities in the standard normal distribution table. The probability for ( Z = -2 ) is approximately 0.0228, and the probability for ( Z = -1 ) is approximately 0.1587.

To find the probability between 49 and 64 years, we subtract the smaller probability from the larger one:

[ P(49 < X < 64) = P(
`Z_(64)`) - P(
`Z_(49)`) ]

[ P(49 < X < 64) = 0.1587 - 0.0228\]

[ P(49 < X < 64) \approx 0.1359 ]

This probability corresponds most closely to option c) 0.4772, making it the correct answer.