## A dishwasher has a mean lifetime of 12 years with an estimated standard deviation of 1.25 years. Assume the lifetime of a dishwasher is norm

Question

A dishwasher has a mean lifetime of 12 years with an estimated standard deviation of 1.25 years. Assume the lifetime of a dishwasher is normally distributed. Round the probabilities to four decimal places. It is possible with rounding for a probability to be 0.0000. a) State the random variable. b) Find the probability that a randomly selected dishwasher has a lifetime of 12.65 years or more. c) Find the probability that a randomly selected dishwasher has a lifetime of 13.55 years or less. d) Find the probability that a randomly selected dishwasher has a lifetime between 12.65 and 13.55 years. e) Find the probability that randomly selected dishwasher has a lifetime that is at most 8.875 years. 0.0062 f) Is a lifetime of 8.875 years unusually low for a randomly selected dishwasher? Why or why not? g) What lifetime do 64% of all dishwashers have less than? Round your answer to two decimal places in the first box. Put the correct units in the second box. years

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3 weeks 2021-10-03T16:15:42+00:00 1 Answer 0

Step-by-step explanation:

Hello!

a. The variable of interest is:

X: Lifetime of a dishwasher. (years)

Assuming this variable has a normal distribution with mean μ= 12 years and a standard deviation of σ= 1.25 years.

To calculate the following probabilities, you have to use the standard normal distribution Z= (X-μ)/δ~N(0;1)

b.

P(X≥12.65)= 1 – P(X<12.65)

1 – P(Z<(12.65-12)/1.25)= 1 – P(Z<0.52)= 1 – 0.698 = 0.302

c.

P(X≤13.55)= P(Z≤(13.55-12)/1.25)= P(Z≤1.24)= 0.893

d.

P(12.65≤X≤13.55)= P(X≤13.55) – P(X≤12.65)= P(Z≤1.24) – P(Z≤0.52)= 0.893-0.698= 0.195

e.

When you are looking for the probability of the variable taking “at most” certain value, this means that you are looking for the probability of it being equal or less to the given value:

P(X≤8.875)= P(Z≤(8.875-12)/1.25)= P(Z≤-2.5)= 0.006

f.

When standardizing the value of X= 8.875 the obtained Z-value was -2.5, this value can be interpreted as 8.875years is -2.5 standard deviations away from the mean. The further value is from its population mean, the lowe is its probability of occurrence, i.e. the more uncommon it is. 2.5 standard deviations is a far enough distance to claim that the value is uncommon.

g.

What lifetime does 64% of all dishwashers have less than?

Symbolically:

P(X≤x₀)= 0.64

x₀ is a value of the variable that has below it 64% of the variable distribution. The first step to finding the value of x₀ is to look for the value under the standard normal distribution that accumulates 0.64 of probability:

P(Z≤z₀)= 0.64

z₀= 0.358

The second step is to reverse the standardization using the values of μ and σ to reach the corresponding value of X.

z₀= (x₀-μ)/σ

x₀= (z₀*σ)+μ

x₀=(0.358*1.25)+12

x₀= 12.4475

The lifetime that at most 64% of the dishwashers is 12.4475 years.

I hope this helps!