## Related questions with answers

A certain type of stainless steel powder is supposed to have a mean particle diameter of $\mu=15 \mu \mathrm{m}.$ A random sample of 87 particles had a mean diameter of $15.2 \mu \mathrm{m},$ with a standard deviation of $1.8 \mu \mathrm{m}.$ A test is made of $H_{0}: \mu=15 \text { versus } H_{1}: \mu \neq 15.$ a. Find the P-value. b. Do you believe it is plausible that the mean diameter is $15 \mu \mathrm{m},$ or are you convinced that it differs from $15 \mu \mathrm{m} ?$ Explain your reasoning.

Solution

VerifiedGiven:

$n=87$

$\overline{x}=15.2$

$\sigma=1.8$

$H_0:\mu=15$

$H_a:\mu\neq 15$

(a) The sampling distribution of the sample mean has mean $\mu$ and standard deviation $\dfrac{\sigma}{\sqrt{n}}$.

The z-value is the sample mean decreased by the population mean, divided by the standard deviation:

$z=\dfrac{\overline{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{15.2-15}{1.8/\sqrt{87}}\approx 1.04$

The P-value is the probability of obtaining a value more extreme or equal to the standardized test statistic $z$, assuming that the null hypothesis is true. Determine the probability using normal probability table in the appendix.

$P=P(Z<-1.04\text{ or }Z>1.04)=2P(Z<-1.04)=2(0.1492)= 0.2984$

(b) The probability in part (a) is not small (larger than 0.05), which indicates that the null hypothesis is not false and thus it is plausible that the mean diameter is 15 $\mu$m.

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