Question

Type the correct answer in the box

A component in a music system has a life expectancy of 2400 hours with a standard deviation of 300 hours. If an average person listens to

music for 1,000 hours in a year, the probability that the component lasts for more than 3 years is

Reset

Ned

2019 Ementum

seserved

Answer 1

If an average person listens to music for 1,000 hours in a year, the probability that the component lasts for more than 3 years is 2.3%.

In Mathematics and Statistics, the z-score of a given sample size or data set can be calculated by using the following formula:

Z-score, z = (x - μ)/σ

Where:

σ represents the standard deviation.

x represents the sample score.

μ represents the mean score.

Since an average person listens to music for 1,000 hours in a year, the total life span of the component for 3 years can be calculated as follows;

Total life span = 3 × 1000

Total life span = 3000 years.

By substituting the parameters into the z-score formula, we have the following:

Z-score, z = (3000 - 2400)/300

Z-score, z = 2.0

Based on the standardized normal distribution table, the required probability is given by:

P(X > 3000) = P(x > Z)

P(X > 3000) = 1 - P(Z < 2)

P(X > 3000) = 1 - 0.9773

Probability = 0.0227 × 100.

Probability = 2.27 ≈ 2.3%.

Answer 2

Answer: 0.0227

Explanation:

Mean expectancy hours M = 2400 hours

Standard deviation S = 300 hours

Number of hours used per year = 1000 hours

Number of hours in 3 years N = 3*1000 = 3000 hours

Probability of lasting more than 3000 hours = 1 - Probability of lasting up to 3000 hours

P(z>3000) = 1 - P(z< 3000)

P(z<3000) = P[(N-M)/S < (3000-2400)/300 ]

P(z<3000) = P(z< 600/300)

P(z<3000) = P(z< 2) = ¢(2)

P(z<3000) = 0.9773

Therefore,

P(z>3000) = 1 - P(z<3000)

P(z>3000) = 1 - 0.9773

P(z>3000) = 0.0227

The probability of the component lasting more than 3 years (3000 hours) is 0.0227