Question

Two brands of cherry pitters were being evaluated for use in a commercial cannery. For machine A, out of 1154 cherries pitted, 1136 were found to be properly pitted. For machine B, 945 out of 963 were pitted properly. We are interested in knowing whether there is a difference between the two machines, with respect to properly pitting cherries.

(a) Check that the assumptions for inference are satisfied.

(b) Is there a difference in the proportion of properly pitted cherries for machines A and B? Use a formal hypothesis test with an α-level of 0.05.

(c) Is there a difference in the proportion of properly pitted cherries for machines A and B? Use a 95% confidence interval to test the hypothesis.

Answer 1

a)
The size of each population is large relative to the sample drawn from the population.

The samples from each population are big enough to justify using a normal distribution to model differences between proportions.

The samples are independent; that is, observations in population 1 are not affected by observations in population 2, and vice versa.

b)

Hypothesis;
H0 ; p1 = p2
Ha: p1 not equals to p2

test statistics:

pcap = ( 1136 + 945)/( 1154+ 963)
= 0.983

p1 = 1136/1154 = 0.984

p2 = 945/963 = 0.981

z = (p1 -p2) / sqrt(pcap *(1-pcap)*(1/n1+1/n2)
= (0.984 - 0.981) / sqrt(0.983 *(1-0.983) * (1/1154 + 1/963))
= 0.5482

p value = 5835

do not reject H0

there is no difference in the proportion of properly pitted cherries for machines A and B

c)

z value at 95% = 1.96

CI = ( p1 - p2) +/- z * sqrt(p1*(1-p1)/n1+ p2 *(1-p2)/n2)
= (0.984 - 0.981) +/- 1.96 * sqrt(0.984 *(1-0.984)/1154 + 0.981 *(1-0.981)/963)
= (-0.0081 , 0.0142 )

There is no difference in the proportion of properly pitted cherries for machines A and B