Try the fastest way to create flashcards
Question

# 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

Verified
Step 1
1 of 2

Given:

$n=87$

$\overline{x}=15.2$

$\sigma=1.8$

$H_0:\mu=15$

$H_a:\mu\neq 15$

(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.

## Recommended textbook solutions #### Statistics for Engineers and Scientists

4th EditionISBN: 9780073401331 (2 more)William Navidi
1,258 solutions #### Engineering Statistics

5th EditionISBN: 9780470905302Douglas C. Montgomery, George C. Runger
778 solutions #### Statistics for Engineers and Scientists

5th EditionISBN: 9781259717604 (2 more)William Navidi
1,294 solutions #### Mühendislik İstatistiği

5th EditionISBN: 9786053556589Douglas C. Montgomery, George C. Runger, Norma F. Hubele
778 solutions