Question

When the parameters of a population are known, the likelihood of obtaining a certain mean value, ¯ x , from a small sample can be predicted by the z‐variable with z = ( ¯ x − x ' ) / σ / √ N . Consider a stamping process in which the applied load has a known true mean of 400 N with standard deviation of 25 N. If 25 measurements of applied load are taken at random, what is the probability that this sampling will have a mean value between 390 and 410?

Answer 1

Answer: 0.9545

Explanation:

Given : Population mean :
`x'=400\ N`

Standard deviation :
`\sigma=25\ N`

Sample size : n=25

Test statistic :
`z=\frac{\overline{x}-x'}{(\sigma)/(√(n))}`

For
`\overline{x}= 390`, we have

`z=(390-400)/((25)/(√(25)))=-2`

For
`\overline{x}= 410`, we have

`z=(410-400)/((25)/(√(25)))=2`

Now, by using the standard normal distribution table for z , we have

`\text{P-value=}P(-2<z<-2)=1-2(P\geq2)\\\\=1-2(1-P(z<2))\\\\=1-2(1-0.9772498)=0.9544996\approx0.9545`

Hence, probability that this sampling will have a mean value between 390 and 410 = 0.9545