The probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.3 inches is 39.54%.
Given mean diameter of 207, variance=9, sample size of 72.
We have to calculate the probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.3 inches.
The sample mean may be greater than or less than from population mean than 0.3 inches.
Either greater than 207+0.3=207.3 inches,
Smaller =207-0.3=206.7
Since the normal distribution is symmetric these probabilities are equal. So we find one of them and multiply by 2.
Probability of being less than 206.7
P value of z when X=206.7. So
Z=(X-μ)/s
=(206.7-207)/0.35
=-0.3/0.35
=-0.857
p value =0.1977
Probability of differing from population mean greater than 0.3 inches=2*0.1977
=0.3954
=39.54%
Hence the probability that the mean diameter of the sample shafts would differ from the population mean by greater than 0.3 inches is 39.54%.
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