Answer to Question #143270 in Statistics and Probability for Auguste Nkole

Question

QUESTION 2 (13 MARKS)

A manufacturer claims that the percentage of faulty items in any lot of the articles he produces is 1%. A random sample of 200 articles is selected and 8 are found to be faulty.

To test the validity of this claim, and using α = 0.01, Answer the following questions

2.1) Formulate the null and alternative hypotheses (2)

2.2) Select and compute the test statistic (4)

2.3) Obtain the critical value and formulate the decision. (4)

2.4) Based on the information obtained in Questions 2.1 ; 2.2 & 2.3, what can you say about the manufacturer’s claim ? (3)

Answer 1

Lets denote p = percentage of faulty items produced and n = 200.

Because of the manufacturer claim:

$H_0:p= 0.01$

$H_1:p\neq0.01$

Significance level $\alpha=0.01$

A Poisson distribution with $\lambda=np$ provides an approximation to a binomial distribution when $n\ge 50$ and $p\le0.1$

Thus X denotes the number of faulted items in the sample, then

$X\sim Po(\lambda=200\cdot0.01=2)$

Using tables of the Cumulative Poisson Distribution Functions

$P(X\ge 8)=1-P(X\le7)=1-0.9989=0.0011$

Since the test is two-tailed this probability is compared with $\frac{\alpha}{2}=0.005$ because a small number of faulted items could also lead to the rejection of the null hypothesis.

Here, $0.0011<0.005$ so $H_0$ is rejected.

There is evidence with a 1% level of significance, that the manufacturer's claim is not valid.