Question

State DMV records indicate that of all vehicles undergoing emissions testing during the previous year, 70% passed on the first try. A random sample of 210 cars tested in a particular county during the current year yields 120 that passed on the initial test. Does this suggest that the true proportion for this county during the current year differs from the previous statewide proportion? Test the relevant hypotheses using a - .05. (a) Choose the appropriate hypotheses. O Hop = 0.70 versus H.:P*0.70 OH : P = 0.70 versus Hp > 0.70 O Hop > 0.70 versus H, p=0.70 O H.:P < 0.70 versus Hp 0.70 O Hop 0.70 versus Hp <0.70 (b) What is the value of the test statistic for this test? (c) What is the critical value for this test? + (d) State your conclusions. O Reject the null hypothesis, there is a significant difference in proportions. Reject the null hypothesis, there is not a significant difference in proportions. Fall to reject the null hypothesis, there is not a significant difference in proportions O Fail to reject the null hypothesis, there is a significant difference in proportions,

Answer 1

Given : n=210 , X=120

The estimate of the sample proportion is ,

$\hat{p}=\frac{X}{n}=\frac{120}{210}=0.5714$

(a) The null and alternative hypothesis is ,

$H_0:p=0.70$ Vs $H_a:p\neq 0.70$

(b) The value of the test statistic is ,

Zstat P - PO Po(1-po) 0.5714 - 0.70 0.70(1-0.70) 210 = -4.07 n

(c) The critical values are ,

; From Z-table

Za a/2 20.05/2 = +1.96

The decision rule is ,

Reject Ho , if Z-stat>1.96 or Z-stat<-1.96

Decision : Here , Z-stat=-4.07<-1.96

Therefore , reject the null hypothesis.

(d) Conclusion : Reject the null hypothesis. there is significant difference in proportions.