Question

A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time is less than 8.9 hours.

Answer 1

Answer: 0.9608

Step-by-step explanation:

Given : A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 Chevrolet Cavalier shows that

Mean :$\mu=8.4\text{ hours}$

Standard deviation : $\sigma=1.8\text{ hours}$

Sample size : $n=40$

Let $\overline{x}$ be the sample mean.

The formula for z-score in a normal distribution :

$z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}$

For  $\overline{x}=8.9$

$z=\dfrac{8.9-8.4}{\dfrac{1.8}{\sqrt{40}}}\approx1.76$

The P-value = $P(\overline{x}<8.9)=P(z<1.76)= 0.960796\approx 0.9608$

Hence, the probability that their mean rebuild time is less than 8.9 hours is 0.9608 .

Answer 2

The probability that their mean rebuild time is less than 8.9 hours is 0.9608

Explanation:

A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time is less than 8.9 hours.

The Chevrolet Cavalier is the line of small cars produced for the model years 1982 until 2005 by Chevrolet. Mechanics is the physics area concerned with the motions of macroscopic objects. The mean is the number average. To calculate we add up all the numbers then divide by how many numbers there are. In other words it is the sum divided by the count.

$\mu= 8.4 hours$

$\sigma = 1.8 hours$

$n=40$

$z = \frac{Xbar-\mu }{\frac{\sigma}{\sqrt{r} } } = \frac{8.9-8.4}{\frac{1.8}{\sqrt{40} } } = 1.76 \\P(Xbar < 8.9) = P(Z <1.76) =0.9608$

The probability of $P\left({Xbar < 8.9}\right)$ is equal to $P\left({Z < 1.76}\right)$.

Therefore the value of $P\left({Z < 1.76}\right)$ from the standard normal table is $0.9608$.

Hence, the probability that their mean rebuild time is less than 8.9 hours is  0.9608  = 96.08%.

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