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

Step-by-step explanation:

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

Standard deviation : $\sigma=25\ N$

Sample size : n=25

Test statistic : $z=\dfrac{\overline{x}-x'}{\dfrac{\sigma}{\sqrt{n}}}$

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

$z=\dfrac{390-400}{\dfrac{25}{\sqrt{25}}}=-2$

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

$z=\dfrac{410-400}{\dfrac{25}{\sqrt{25}}}=2$

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

$\text{P-value=}P(-2$

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