Question

A machine used to fill​ gallon-sized paint cans is regulated so that the amount of paint dispensed has a mean of 124 ounces and a standard deviation of 0.40 ounce. You randomly select 40 cans and carefully measure the contents. The sample mean of the cans is 123.9 ounces. Does the machine need to be​ reset? Explain your reasoning.

(Yes/No)______, it is (likely/ very unlikely) _____ that you would have randomly sampled 40 cans with a mean equal to 123.9 ounces, because it (does not lie/ lies) ______ within the range of a usual event, namely within (1 deviation/ 2 deviations/ 3 deviations) ______ of the mean of the sampled means.

Please explain your answers! Thanks!

Answer 1

the PDF of normal distribution is = 1/σ * √2π * e ^ -(x-u)^2/ 2σ^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd/sqrt(n) ~ N(0,1)
mean of the sampling distribution ( x ) = 124
standard Deviation ( sd )= 0.4/ Sqrt ( 40 ) =0.0632
sample size (n) = 40
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for sample mean, P(X = 123.9) = (123.9-124)/0.4/ Sqrt ( 40 )
= -0.1/0.0632= -1.5811
Z = -1.5811
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68% OF DATA
About 68% of the area under the normal curve is within one standard deviation of the mean. i.e. (u ± 1s.d)
So to the given normal distribution about 68% of the observations lie in between
= [0 ± 1]
= [ 0 - 1 , 0 + 1]
= [ -1 , 1 ]
95% OF DATA
About 95% of the area under the normal curve is within two standard deviation of the mean. i.e. (u ± 2s.d)
So to the given normal distribution about 95% of the observations lie in between
= [0 ± 2 * 1]
= [ 0 - 2 * 1 , 0 + 2* 1]
= [ -2 , 2 ]
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The value of z-score is –1.58, this value lies within the two standard deviations from the mean. This indicates that -1.58 is not unusual but it is likely.
It is not necessary that the machine has to be reset as, the z score value of randomly sampled 40 cans with a mean equal to 125.9 ounces is likely and lies within 2 standard deviations of the mean.

No, it is likely that a randomly sampled 40 cans with a mean equal to 123.9 ounces because it lies within 2 standard deviations of the mean of the sample means.