3. Replacement times for CD players are normally distributed with a mean of seve years and the variance of two. Find the replacement time separating the bottom 259 to the top 75%.



Answer :

Answer:

Approximately 6.05 years

Step-by-step explanation:

Use the properties of the normal distribution

Given:

Mean() = 7 years

Variance (^2) = [tex]2 years^{2}[/tex]

Standard deviation() = [tex]\sqrt{2}[/tex] years

Find the value of x such that 25% of the distribution lies below x and 75% lies above x.

The z-score z for the 25th percentile is approximately -0.674.

The relationship between the z-score and the value x in the normal distribution is given by the formula:

x = μ + z ⋅ σ

Substitute the given values

x = 7 + (-0.674) * [tex]\sqrt{2}[/tex]

Calculate for x

[tex]\sqrt{2}[/tex]    = approx 1.414

x = 7 + (-0.674) * 1.414

x = 7 - 0.953

x = approx 6.047

Therefore, the replacement time separating the bottom 25% from the top 75% is approximately 6.05 years.