Question

question 9 - of 13 step 4 of 6
an engineer has designed a valve that will regulate water pressure on an automobile engine. the valve was tested on 180 engines and the mean pressure was 5.4 pounds/square inch (psi). assume the population standard deviation is 0.9. if the valve was designed to produce a mean pressure of 5.2 psi, is there sufficient evidence at the 0.05 level that the valve performs above the specifications?
step 4 of 6: find the p - value of the test statistic. round your answer to four decimal places.
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Explanation:

Step1: Calculate the test - statistic

The formula for the z - test statistic in a one - sample z - test is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean under the null hypothesis, $\sigma$ is the population standard deviation, and $n$ is the sample size.
Given $\bar{x} = 5.4$, $\mu=5.2$, $\sigma = 0.9$, and $n = 180$.
$z=\frac{5.4 - 5.2}{\frac{0.9}{\sqrt{180}}}=\frac{0.2}{\frac{0.9}{13.4164}}=\frac{0.2}{0.0671}=2.98$.

Step2: Find the P - value

Since this is a right - tailed test (we want to know if the valve performs above the specifications), the P - value is $P(Z>z)$.
Using the standard normal distribution table, $P(Z > 2.98)=1 - P(Z\leq2.98)$.
From the standard normal table, $P(Z\leq2.98)=0.9986$.
So, $P(Z > 2.98)=1 - 0.9986 = 0.0014$.

Answer:

$0.0014$