Question

a quality - control manager at a widget factory knows that historically a proportion 0.068 of all widgets fail inspection. she wants to check whether the new assembly line has changed that failure rate. she takes a random sample of 185 widgets off the line and finds 16 defects.
perform a hypothesis test with a significance level of 0.10 using the p - value method to test if changing the assembly line has changed the failure rate.
a. choose the appropriate null and alternative hypotheses.
a. ( h_0:p = 0.068,h_a:p
eq0.068 )
b. ( h_0:p = 0.068,h_a:plt0.068 )
c. ( h_0:p = 0.068,h_a:pgt0.068 )
d. ( h_0:pgt0.068,h_a:p = 0.068 )
e. ( h_0:plt0.068,h_a:p = 0.068 )
b. compute the test statistic and associated p - value. (round each value as instructed.)
( z=) 1.39 (round your answer to two decimal places.)
( p - value=) 0.0466 (round your answer to four decimal places.)
c. what is the correct conclusion?
a. we should not reject ( h_0 ). there is sufficient evidence to conclude that the failure rate is different.
b. we should reject ( h_0 ). there is sufficient evidence to conclude that the failure rate is different.
c. we should reject ( h_0 ). there is insufficient evidence to conclude that the failure rate is different.
d. we should not reject ( h_0 ). there is insufficient evidence to conclude that the failure rate is different.

Explanation:

Step1: Hypothesis Testing Basics

In hypothesis testing, if the \( P - value\leq\alpha\) (significance level), we reject the null hypothesis \(H_0\). Here, \(\alpha = 0.10\) and \(P - value=0.0466\).

Step2: Decision Rule Application

Since \(0.0466<0.10\) (i.e., \(P - value<\alpha\)), we reject the null hypothesis \(H_0:p = 0.068\). Rejecting \(H_0\) means there is sufficient evidence to conclude that the failure rate is different.

Answer:

B. We should reject \(H_0\). There is sufficient evidence to conclude that the failure rate is different.