Question

A manufacturer of apple phone charges claims that only 4% of his production is defective. In a random sample of 120 items, the defect rate is 4.99%. At the 1% level of significance, test the manager's claim.

i. State the null and alternative hypotheses for the test. [2]

ii. Calculate the value of the test statistic for this test. [2]

iii. Determine the critical region(s) for this test. [2]

iv. State the conclusion of this test. Give a reason for your answer. [3]

Answer 1

$i:\\H_0:p=0.04\\H_1:p>0.04 \left( if\,\,it\,\,is\,\,less, it\,\,is\,\,ok\,\,for\,\,the\,\,manager \right) \\ii:\\Z=\sqrt{n}\frac{\hat{p}-p}{\sqrt{p\left( 1-p \right)}}=\sqrt{120}\frac{0.0499-0.04}{\sqrt{0.04\left( 1-0.04 \right)}}=0.553427\\iii:\\\left\{ Z>z_{0.99} \right\} =\left\{ Z>2.236 \right\} \,\,is\,\,a\,\,critical\,\,region\\iv:\\Since\,\,the\,\,value\,\,of\,\,statistic\,\,is\,\,not\,\,in\,\,the\,\,critical\,\,region, the\,\,null\,\,hypothesis\,\,is\,\,not\,\,declined.$