Recent incidents of food contamination have caused great concern among consumers
ID: 3313902 • Letter: R
Question
Recent incidents of food contamination have caused great concern among consumers. An article reported that 39 of 80 randomly selected Brand A brand chickens tested positively for either campylobacter or salmonella (or both), the leading bacterial causes of food-borne disease, whereas 67 of 80 Brand B brand chickens tested positive. (a) Does it appear that the true proportion of non-contaminated Brand A chickens differs from that for Brand B? Carry out a test of hypotheses using a significance level 0.01. (Use pi for Brand A and pz for Brand B.) State the relevant hypotheses. O Ho: P1-P2 = 0 Ha: P1 - P2 + 0 O Ho: P1 - P2 = 0 Ha: P1 - P2 0 Hai P1 - P2 = 0 O Ho: P1 - P2 0 Calculate the test statistic and P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = State the conclusion in the problem context. O Fail to reject Ho. The data suggests the true proportion of non-contaminated chickens differs for the two companies. O Reject Ho. The data does not suggest the true proportion of non-contaminated chickens differs for the two companies. O Fail to reject Ho. The data does not suggest that the true proportion of non-contaminated chickens differs for the two companies. O Reject Ho. The data suggests that the true proportion of non-contaminated chickens differs for the two companies. (b) If the true proportions of non-contaminated chickens for the Brand A and Brand B are 0.50 and 0.25, respectively, how likely is it that the null hypothesis of equal proportions will be rejected when a 0.01 significance level is used and the sample sizes are both 100? (Round your answer to four decimal places.)Explanation / Answer
Given that,
sample one, x1 =39, n1 =80, p1= x1/n1=0.488
sample two, x2 =67, n2 =80, p2= x2/n2=0.838
null, Ho: p1 = p2
alternate, H1: p1 != p2
level of significance, = 0.01
from standard normal table, two tailed z /2 =2.576
since our test is two-tailed
reject Ho, if zo < -2.576 OR if zo > 2.576
we use test statistic (z) = (p1-p2)/(p^q^(1/n1+1/n2))
zo =(0.488-0.838)/sqrt((0.663*0.338(1/80+1/80))
zo =-4.681
| zo | =4.681
critical value
the value of |z | at los 0.01% is 2.576
we got |zo| =4.681 & | z | =2.576
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -4.6813 ) = 0
hence value of p0.01 > 0,here we reject Ho
ANSWERS
---------------
null, Ho: p1 - p2 = 0
alternate, H1: p1 - p2 != 0
test statistic: -4.68
critical value: -2.576 , 2.576
decision: reject Ho, the data suggest that the true proportion of non contamminated
chicken differs for the 2 companies
p-value: 0
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