Question

The shape of the distribution of the time required to get an oil change at a 10​-minute oil-change facility is unknown. However, records indicate that the mean time is 11.8 minutes​, and the standard deviation is 4.9 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 sample size needs to be greater than or equal to 30.

D. The normal model cannot be used if the shape of the distribution is unknown.

​(b) What is the probability that a random sample of n=35 oil changes results in a sample mean time less than 10 minutes?​

​(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 35 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 blank minutes.

Answer 1

(a) To compute probability regarding the sampling mean using the normal model, the size of sample that would be required is

(C) The sample size needs to be greater than or equal to 30.

[ This is because by Central limit theorem , for a population with mean and standard deviation which take sufficiently large random sample ( 30) from the population, the distribution of sample means will be approximately normal .]

(b) n= sample size = 35

sample mean =

population mean =

population standard deviation =

Since by central limit theorem for sufficiently large n ( here 35),

probability that a random sample of n=35 oil changes results in a sample mean time less than 10 minutes

=

Here,

= 1-0.9849 =0.0151

Therefore,  probability that a random sample of n=35 oil changes results in a sample mean time less than 10 minutes is 0.0151

(D) Let there is a 10% chance of being at or below a mean oil change time of 'a' minutes.

Here n = 35

Here,

or, a = 10.73

ANS : There is a 10% chance of being at or below a mean oil change time of 10.73 minutes.

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