are given in the following table.Does this study provide statistical evidence at the α = 0.01 level that the vaccines differ in their
effectiveness?
solution:I . Let p 1 = the population proportion infected after receiving Vaccine A.
Let p 2 = the population proportion infected after receiving Vaccine B.
H 0 : p 1 – p 2 = 0.H (^) A : p 1 – p 2 ≠ 0.
II . We will use a two-proportion z -test at α = 0.01.
n 1 1 = 225(0.453), = 102, n 1 (1 – 1 ) = 225(0.547) = 123,
n 2 2 = 285(0.333) = 95 n 2 = 2 = 225(0.667) = 190.
All values are larger than 5, so the conditions necessary are present for the two-proportion z
-test.
(Note: When the counts are given in table form, as in this exercise, the values found in the
calculation above are simply the table entries! Take a look. Being aware of this could save
you some work.)
III .
0.006 (from Table A). (Remember that you have to multiply by 2 since it is a two-sided
alternative—you are actually finding the probability of being 2.76 standard deviations away
from the mean in some direction.) Given the z -score, the P -value could also be found using
a TI-83/84: 2 × normalcdf (2.76,100).(Using a TI-83/84, all the mechanics of the exercise could have been done using the 2-
PropZTest in the STAT TESTS menu. If you do that, remember to show enough work that
someone reading your solution can see where the numbers came from.)