Question

Suppose Deidre, a quality assurance specialist at a lab equipment company, wants to determine whether or not the company's two primary manufacturing centers produce test tubes with the same defect rate. She suspects that the proportion of defective test tubes produced at Center A is less than the proportion at Center B.

Deidre plans to run a -
test of the difference of two proportions to test the null hypothesis, 0:=
, against the alternative hypothesis, :<
, where
represents the proportion of defective test tubes produced by Center A and
represents the proportion of defective test tubes produced by Center B. Deidre sets the significance level for her test at =0.05
. She randomly selects 535 test tubes from Center A and 466 test tubes from Center B. She has a quality control inspector examine the items for defects and finds that 14 items from Center A are defective and 22 items from Center B are defective.



Compute the -
statistic for Deidre's -
test of the difference of two proportions, −
.

Answer 1

The formula to calculate the test statistic for Deidre's test of the difference of two proportions is:

z = (p1 - p2) / sqrt(p * (1 - p) * (1/n1 + 1/n2))

where:
p1 = 14/535 = 0.0262 (proportion of defective test tubes produced by Center A)
p2 = 22/466 = 0.0471 (proportion of defective test tubes produced by Center B)
p = (14 + 22) / (535 + 466) = 0.0343 (pooled proportion)
n1 = 535 (sample size from Center A)
n2 = 466 (sample size from Center B)

Substituting the values, we get:

z = (0.0262 - 0.0471) / sqrt(0.0343 * (1 - 0.0343) * (1/535 + 1/466)) = -2.32

Therefore, the test statistic for Deidre's test of the difference of two proportions is -2.32.

Answer 2

The statistic for Deidre's test of the difference of two proportions is -1.74.