An engineer has designed a valve that will regulate water pressure on an automobile engine. The valve was tested on 150 engines and the mean pressure was 7.7 lbs/square inch. Assume the standard deviation is known to be 0.5. If the valve was designed to produce a mean pressure of 7.6 lbs/square inch, is there sufficient evidence at the 0.1 level that the valve does not perform to the specifications? State the null and alternative hypotheses for the above scenario.

Answer:We are given that The valve was tested on 150 engines and the mean pressure was 7.7 lbs/square inch.[tex]\bar{x}=7.7\\n = 150 \\\sigma = 0.5[/tex]We are also given that the valve was designed to produce a mean pressure of 7.6 lbs/square inchSo, [tex]\mu = 7.6[/tex]Null hypothesis: [tex]H_0:\mu = 7.6[/tex]Alternate hypothesis :  [tex]H_a:\mu \neq 7.6[/tex]Since n > 30 and population standard deviation is given So, We will use z testFormula : [tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]Substitute the values [tex]z=\frac{7.7-7.6}{\frac{0.5}{\sqrt{150}}}[/tex] [tex]z=2.449[/tex]refer the z table for p value so, p value is 0.9927Since it is a two tailed test So, p = 2(1-  0.9927) = 0.0146α = 0.1p value< αSo, we reject null hypothesis Hence There is  sufficient evidence at the 0.1 level that the valve does not perform to the specifications