Question

the shape of the distribution of the time required to get an oil change at a 10 - minute oil - change facility is skewed right. however, records indicate that the mean time is 11.1 minutes, and the standard deviation is 3.3 minutes. complete parts (a) through (c).
(a) to compute probabilities regarding the sample mean using the normal model, what size sample would be required?
a. the sample size needs to be less than or equal to 30.
b. any sample size could be used.
c. the normal model cannot be used if the shape of the distribution is skewed right.
d. the sample size needs to be greater than or equal to 30.
(b) what is the probability that a random sample of ( n = 45 ) oil changes results in a sample mean time less than 10 minutes?
the probability is approximately 0.0125
(round to four decimal places as needed.)
(c) suppose the manager agrees to pay each employee a $50 bonus if they meet a certain goal. on a typical saturday, the oil - change facility will perform 45 oil changes between 10 a.m. and 12 p.m. treating this as a random sample, there would be a 10% chance of the mean oil - change time being at or below what value? this will be the goal established by the manager.
there is a 10% chance of being at or below a mean oil - change time of ( square ) minutes
(round to one decimal place as needed.)

Explanation:

Step1: Find the z - score corresponding to the 10th percentile

The z - score \(z\) for the 10th percentile (\(P(X\leq x)=0.10\)) can be found using a standard normal table or a calculator. Using a standard normal table or a calculator (e.g., in R: qnorm(0.10) or in Excel: NORM.S.INV(0.10)), we get \(z=- 1.28\).

Step2: Use the formula for the sample mean in the normal distribution

The formula for the sample mean \(\bar{x}\) in the normal distribution is \(\bar{x}=\mu+z\frac{\sigma}{\sqrt{n}}\).
We know that \(\mu = 11.1\) (population mean), \(\sigma = 3.3\) (population standard deviation), \(n = 45\) (sample size), and \(z=-1.28\).
First, calculate \(\frac{\sigma}{\sqrt{n}}=\frac{3.3}{\sqrt{45}}\approx\frac{3.3}{6.7082}\approx0.492\).
Then, \(\bar{x}=11.1+(-1.28)\times0.492\).

$$
LATEXBLOCK0
$$

Answer:

\(10.5\)