In the following questions, carry out the procedures following the required form
ID: 3315513 • Letter: I
Question
In the following questions, carry out the procedures following the required format, and for the tests, be sure to calculate the P-value.
Please do a, b
2. In a test on the quality of two television commercials (labeled A, and B), each commercial was shown in a separate area several times over a one-week period. The following week a telephone survey was conducted to identify individuals who had seen the commercials. These individuals were asked if they could recall the primary message in the commercial. Of the 200 individuals that had viewed commercial A, 60 could recall the primary message in the commercial. For commercial B, 63 individuals out of 150 viewers could recall the primary message.
a) Estimate with 90% confidence the difference in recall proportions for the two commercials?
b) How many individuals would need to be sampled in each area, to estimate the difference in proportions to within a margin of error of 5%, using 90% confidence? [Assume equal sample sizes]
Explanation / Answer
2.a.
TRADITIONAL METHOD
given that,
sample one, x1 =60, n1 =200, p1= x1/n1=0.3
sample two, x2 =63, n2 =150, p2= x2/n2=0.42
I.
standard error = sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where
p1, p2 = proportion of both sample observation
n1, n2 = sample size
standard error = sqrt( (0.3*0.7/200) +(0.42 * 0.58/150))
=0.052
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.1
from standard normal table, two tailed z /2 =1.64
margin of error = 1.64 * 0.052
=0.085
III.
CI = (p1-p2) ± margin of error
confidence interval = [ (0.3-0.42) ±0.085]
= [ -0.205 , -0.035]
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DIRECT METHOD
given that,
sample one, x1 =60, n1 =200, p1= x1/n1=0.3
sample two, x2 =63, n2 =150, p2= x2/n2=0.42
CI = (p1-p2) ± sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where,
p1, p2 = proportion of both sample observation
n1,n2 = size of both group
a = 1 - (confidence Level/100)
Za/2 = Z-table value
CI = confidence interval
CI = [ (0.3-0.42) ± 1.64 * 0.052]
= [ -0.205 , -0.035 ]
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interpretations:
1) we are 90% sure that the interval [ -0.205 , -0.035] contains the difference between
true population proportion P1-P2
2) if a large number of samples are collected, and a confidence interval is created
for each sample, 90% of these intervals will contains the difference between
true population mean P1-P2
b.
Compute Sample Size ( n ) = n=(Z/E)^2*p*(1-p)
Z a/2 at 0.1 is = 1.645
Sample Proportion = 0.42
ME = 0.05
n2 = ( 1.645 / 0.05 )^2 * 0.42*0.58
= 263.675 ~ 264
Compute Sample Size ( n ) = n=(Z/E)^2*p*(1-p)
Z a/2 at 0.1 is = 1.645
Sample Proportion = 0.3
ME = 0.05
n1= ( 1.645 / 0.05 )^2 * 0.3*0.7
= 227.306 ~ 228
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