Question

A study of the amount of time it takes a mechanic to rebuild the transmission for a 2010 chevrolet colorado 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. round to four decimal places.

Answer 1

solution: we have, mean =8.4 hrs, std. deviation = 1.8 hrs, sample size n = 40 , X = 8.9 Probability(X<8.9) = ? we know that, Z = (X - mean)/(std. deviation/(sqrt. n)) = (8.9 - 8.4)/(1.8/(sqrt.40)) Z = 1.7568 from standard normal probabilities table, we have , P(Z<1.7568) = 0.9608 Hence, probability that the 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 $\boxed{{\text{probability}} = {\text{0}}{\text{.9608}}}$ or $\boxed{{\text{probability}} = 96.08\% }$.

Further Explanation:

The Z score of the standard normal distribution can be obtained as,

$\boxed{{\text{Z}} = \frac{{X - \mu }}{{\frac{\sigma }{{\sqrt n }}}}}$

Given:

The mean of time is $\boxed{\mu  = 8.4{\text{ hours}}}$

The standard deviation of the time is $\boxed{1.8 hours}$.

The number of mechanics is $\boxed{n = 40}$.

Explanation:

The probability that their mean rebuild time is less than 8.9 can be expressed as $P\left({X < 8.9}\right)$.

The value of standard normal Z can be obtained as follows.

$\begin{aligned}Z&=\frac{{X - \mu }}{{\frac{\sigma }{{\sqrt n}}}}\\&=\frac{{8.9 - 8.4}}{{\frac{{1.8}}{{\sqrt{40}}}}}\\&=\frac{{3.16}}{{1.8}}\\&=1.76\\\end{aligned}$

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

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 $\boxed{{\text{probability}} = {\text{0}}{\text{.9608}}}$ or $\boxed{{\text{probability}} = 96.08\% }$.

Learn more:

1. Learn more about normal distribution https://brainly.com/question/12698949

2. Learn more about standard normal distribution https://brainly.com/question/13006989

Answer details:

Grade: College

Subject: Statistics

Chapter: Normal distribution

Keywords: Z-score, Z-value, standard normal distribution, standard deviation, test, measure, probability, low score, mean, normal distribution, percentile, percentage, undesirable behavior, proportion, mechanic, Chevrolet Colorado, rebuild, transmission, study.