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aleksandr82 [10.1K]
3 years ago
14

An engineer designed a valve that will regulate water pressure on an automobile engine. The engineer designed the valve such tha

t it would produce a mean pressure of 5.85.8 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 290290 engines and the mean pressure was 5.95.9 pounds/square inch. Assume the variance is known to be 0.640.64. A level of significance of 0.010.01 will be used. Make a decision to reject or fail to reject the null hypothesis.
Mathematics
1 answer:
iren [92.7K]3 years ago
7 0
<h2>Answer with explanation:</h2>

Let \mu be the population mean.

By considering the given information , we have

Null hypothesis : H_0: \mu=5.85

Alternative hypothesis :  H_0: \mu>5.85

Since alternative hypothesis is right-tailed , so the test is a right-tailed test.

Given : n=290  ; \overline{x}=5.95  ;

\sigma=\sqrt{Var}=\sqrt{0.64}=0.8

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

i.e. z=\dfrac{5.95-5.85}{\dfrac{0.8}{\sqrt{290}}}\approx2.13

By using the standard normal distribution table for z , we have

P-value for right tailed test : P(Z>z)=1-P(Z

=1-P(z

Since , the P-value is greater than the significance level of \alpha=0.01 , it means we do not have evidence to reject the null hypothesis.

Hence, we conclude that we do not have enough evidence to support the claim that valve performs above the specifications.

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Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : 

                     x^2-(16/25)=0 

Step by step solution :<span>Step  1  :</span> 16 Simplify —— 25 <span>Equation at the end of step  1  :</span><span><span> 16 (x2) - —— = 0 25 </span><span> Step  2  :</span></span>Rewriting the whole as an Equivalent Fraction :

<span> 2.1 </span>  Subtracting a fraction from a whole 

Rewrite the whole as a fraction using <span> 25 </span> as the denominator :

<span> x2 x2 • 25 x2 = —— = ——————— 1 25 </span>

<span>Equivalent fraction : </span>The fraction thus generated looks different but has the same value as the whole 

<span>Common denominator : </span>The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

<span> 2.2 </span>      Adding up the two equivalent fractions 
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

<span> x2 • 25 - (16) 25x2 - 16 —————————————— = ————————— 25 25 </span>Trying to factor as a Difference of Squares :

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</span>Proof :<span>  (A+B) • (A-B) =
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</span>Note : <span> <span>AB = BA </span></span>is the commutative property of multiplication. 

Note : <span> <span>- AB + AB </span></span>equals zero and is therefore eliminated from the expression.

Check :  25  is the square of  5 
Check : 16 is the square of 4
Check : <span> x2  </span>is the square of <span> x1 </span>

Factorization is :       (5x + 4)  •  (5x - 4) 

<span>Equation at the end of step  2  :</span> (5x + 4) • (5x - 4) ——————————————————— = 0 25 <span>Step  3  :</span>When a fraction equals zero :<span><span> 3.1 </span>   When a fraction equals zero ...</span>

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the <span>denominator, </span>Tiger multiplys both sides of the equation by the denominator.

Here's how:

(5x+4)•(5x-4) ————————————— • 25 = 0 • 25 25

Now, on the left hand side, the <span> 25 </span> cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
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Theory - Roots of a product :

<span> 3.2 </span>   A product of several terms equals zero.<span> 

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Solving a Single Variable Equation :

<span> 3.3 </span>     Solve  :    5x+4 = 0<span> 

 </span>Subtract  4  from both sides of the equation :<span> 
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<span> 3.4 </span>     Solve  :    5x-4 = 0<span> 

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<span><span> x = 4/5 = 0.800
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