Question

In manufacturing processes, it is of interest to know with confidence the proportion of defective parts. Suppose that we want to be reasonably certain that less than 4% of a company’s widgets are defective. To test this, we obtain a random sample of 250 widgets from a large batch. Each of the 250 widgets is tested for defects, and 6 are determined to be defective, based upon the manufacturer’s standards. Using α = 0.01, is this evidence that less than 4% of the company’s widgets are defective? State the hypotheses, list and check the conditions, calculate the test statistic, find the p-value, and make a conclusion in a complete sentence related to the scenario.

Answer 1

Answer:

The calculated value Z =   1.3114 < 2.576 at 0.01 level of significance

null hypothesis is accepted

The  evidence that less than 4% of the company’s widgets are defective

Step-by-step explanation:

Step(i):-

Given Population proportion

                                           P = 4% = 0.04

Given sample proportion

                                      $p = \frac{6}{250} = 0.024$

Null hypothesis :- P < 0.04

Alternative Hypothesis :  P ≠ 0.04

Step(ii):-

Test statistic

                    $Z = \frac{p-P}{\sqrt{\frac{PQ}{n} } }$

                    $Z = \frac{0.024-0.04}{\sqrt{\frac{0.04 X 0.96}{250} } }$

                   Z = - 1.3114

              |Z| = | - 1.3114|

The tabulated value Z = 2.576

The calculated value Z =   1.3114 < 2.576 at 0.01 level of significance

null hypothesis is accepted

The  evidence that less than 4% of the company’s widgets are defective