Answer:
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!