A random sample of 9th grade math students was asked if they prefer working thei

Question

A random sample of 9th grade math students was asked if they prefer working their math problems using a pencil or a pen. Of the 250 students surveyed, 100 preferred pencil and 150 preferred pen.

(a) Using the results of this survey, construct a 95 percent confidence interval for the proportion of 9th grade students that prefer to work their math problems in pen.

(b) A school newspaper reported on the results of this survey by saying, “Over half of ninth-grade math students prefer to use pen on their math assignments.” Is this statement supported by your confidence interval? Explain.

Explanation / Answer

TRADITIONAL METHOD
given that,
possibile chances (x)=150
sample size(n)=250
success rate ( p )= x/n = 0.6
I.
sample proportion = 0.6
standard error = Sqrt ( (0.6*0.4) /250) )
= 0.031
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
margin of error = 1.96 * 0.031
= 0.061
III.
CI = [ p ± margin of error ]
confidence interval = [0.6 ± 0.061]
= [ 0.539 , 0.661]
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DIRECT METHOD
given that,
possibile chances (x)=150
sample size(n)=250
success rate ( p )= x/n = 0.6
CI = confidence interval
confidence interval = [ 0.6 ± 1.96 * Sqrt ( (0.6*0.4) /250) ) ]
= [0.6 - 1.96 * Sqrt ( (0.6*0.4) /250) , 0.6 + 1.96 * Sqrt ( (0.6*0.4) /250) ]
= [0.539 , 0.661]
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interpretations:
1. We are 95% sure that the interval [ 0.539 , 0.661] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population proportion

a.
95% confidence interval for the proportion of 9th grade students that prefer to work their math problems in pen
[0.539 , 0.661] = 53.9%,66.1%

b.
yes,
Over half of ninth-grade math students prefer to use pen on their math assignments.
confidence interval is [0.539 , 0.661]

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