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To test the ability of mechanics to identify simple engine problems, a car with a single such problem was taken in turn to 72 different car repair facilities. Only 42 of the 72 mechanics who worked on the car correctly identified the problem. Does this strongly indicate that the true proportion of mechanics who could identify this problem is less than 0.75?


a) Perform the appropriate test of hypothesis using = 0.01. Clearly show the eight (8) steps



Answer :

Using the z-distribution, it is found that since the p-value is less than 0.01, there is strong indication that the true proportion of mechanics who could identify this problem is less than 0.75.

What are the hypotheses tested?

At the null hypotheses, it is tested if there is not enough evidence that the proportion is less than 0.75, that is:

[tex]H_0: p \geq 0.75[/tex]

At the alternative hypotheses, it is tested if there is enough evidence that the proportion is less than 0.75, that is:

[tex]H_1: p < 0.75[/tex]

What is the test statistic?

The test statistic is given by:

[tex]z = \frac{\overline{p} - p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]

In which:

  • [tex]\overline{p}[/tex] is the sample proportion.
  • p is the proportion tested at the null hypothesis.
  • n is the sample size.

For this problem, the parameters are given as follows:

[tex]n = 72, \overline{p} = \frac{42}{72} = 0.5833, p = 0.75[/tex]

Hence the test statistic is given by:

[tex]z = \frac{\overline{p} - p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]

[tex]z = \frac{0.5833 - 0.75}{\sqrt{\frac{0.75(0.25)}{72}}}[/tex]

z = -3.27.

What is the p-value of the test?

Considering a left-tailed test, as we are testing if the proportion is less than a value, with z = -3.27, the p-value is < 0.01.

Since the p-value is less than 0.01, there is strong indication that the true proportion of mechanics who could identify this problem is less than 0.75.

More can be learned about the z-distribution at https://brainly.com/question/13873630

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