In: Math
Researchers at Consumer Reports recently found that fish are often mislabeled in grocery stores and restaurants. They are interested to know, however, if the proportion of mislabeling varies by type of fish. They collected data on 400 packages of tuna and 300 packages of mahi mahi and found that 110 and 95 were mislabeled, respectively.
a. What are the point estimates for the proportion of tuna and mahi mahi that are mislabeled?
b. Provide a 95% confidence interval estimate of the difference between the proportion of tuna and mahi mahi that is mislabeled.
c. Based on your answer to (b), would you say the rate of mislabeling is different for tuna and mahi mahi? Explain your answer.
d. Now, let’s say we want to test whether the proportion of tuna mislabeled is lower than the proportion of mislabeled mahi mahi. Assuming a 99% confidence level, work through your hypothesis testing procedure below.
a) point estimate of population proportion is sample proportion
Let p1 and p2 are proportions of mislabeled tuna and mahi mahi
b) 95% confidence interval of difference between proportions of mislabeled tuna and mahi mahi is given by
where
P= (110+95)/ (400+300) = 0.29
Q=1-P = 0.71
For 95% confidence , zc =1.96
Therefore
95% CI is
=
= (-0.11 , 0.03 )
(c) Since the confidence interval includes 0 ( lower limit is negative and upper limit is positive)
There is not sufficient evidence to conclude that rate of misleading is different for tuna and mahi mahi
(d)
The null and alternative hypothesis
H0: p1=p2
Ha: p1<p2
Test statistic
= - 1.20
The left tailed critical value of z at 99% confidence ()
zc =- 2.33 (from z table)
Critical region , z < -2.33
Since calculated z =-1.20 is not in the critical region
We fail to reject H0
There is not sufficient evidence to conclude that proportion of tuna mislabeled is less than proportion of mislabeled mahi mahi