Question

A manufacturer of paper coffee cups would like to estimate the proportion of cups that are defective (tears, broken seems, etc.) from a large batch of cups. They take a random sample of 200 cups from the batch of a few thousand cups and found 18 to be defective. The goal is to perform a hypothesis test to determine if the proportion of defective cups made by this machine is more than 8%.
Required:
a. Calculate a 95% confidence interval for the true proportion of defective cups made by this machine.
b. What is the sample proportion?
c. What is the critical value for this problem?
d. What is the standard error for this problem?

Answer 1

Answer:

a

  The 95% confidence interval is  $0.0503 < p < 0.1297$

b

The sample proportion is  $\r p = 0.09$

c

The critical value is  $Z_{\frac{\alpha }{2} } = 1.96$

d

 The standard error is  $SE =0.020$

Step-by-step explanation:

From the question we are told that

   The  sample size is  n =  200

     The number of defective is  k =  18

The null hypothesis is  $H_o : p = 0.08$

The  alternative hypothesis is  $H_a : p > 0.08$

Generally the sample proportion is mathematically evaluated as

            $\r p = \frac{18}{200}$

            $\r p = 0.09$

Given that the confidence level is  95% then the level  of significance is mathematically evaluated as

        $\alpha = 100 - 95$

        $\alpha = 5\%$

        $\alpha = 0.05$

Next we obtain the critical value of  $\frac{ \alpha }{2}$ from the normal distribution table, the value is  

        $Z_{\frac{\alpha }{2} } = 1.96$

Generally the standard of error is mathematically represented as

          $SE = \sqrt{\frac{\r p (1 - \r p)}{n} }$

substituting values

         $SE = \sqrt{\frac{0.09 (1 - 0.09)}{200} }$

        $SE =0.020$

The  margin of error is  

       $E = Z_{\frac{ \alpha }{2} } * SE$

=>    $E = 1.96 * 0.020$

=>   $E = 0.0397$

The  95% confidence interval is mathematically represented as

     $\r p - E < \mu < p < \r p + E$

=>   $0.09 - 0.0397 < \mu < p < 0.09 + 0.0397$

=>  $0.0503 < p < 0.1297$