Question

A random sample of 200 freshman and 100 seniors at
Ferris High School are asked whether they agree with a plan to
excuse upper class students (juniors and seniors) a half hour early
while keeping underclass students (freshmen and sophomores) in
school for the last half hour. Of the students sampled, 160
freshmen opposed the plan and 20 seniors opposed the plan. Is there a difference (5% level of significance) between the proportion of freshmen who oppose the plan and the proportion of seniors who oppose it? Do a complete test showing all the parts. |

Answer #1

Null hypotheses
H_{o} : There is no difference between the
proportion of freshman who opposes plan and the proportion of
seniors who oppose it .

Alternative hypotheses
H_{1} : There is a difference between the proportion
of freshman who opposes plan and the proportion of seniors who
oppose it.

Test statistic = Z = ( p1 - p2 ) / (( p*(1-p) ( (1/n1) + (1/n2)
))^{0.5}

p1 = 160/200 = 0.8

p2 = 20 / 100 =0.2

p = ( 0.8*n_{1} + 0.2* n_{2} ) / (n_{1}
+ n_{2} )

n_{1} = 200 and n_{2} = 100

p = (0.8 * 200 + 0.2*100 )/ ( 300) = 0.6

Z= ( 0.8 - 0.2 ) / ( 0.6 * 0.4 * ( (1/200) +
(1/100)))^{0.5}

Z = 10

tabulated Z = Z_{0.025} = **1.96**

Since, Calculated Z = 10 > 1.96 (tabulated Z) , we reject Ho
at 5 % level of significance and conclude that **there is a
difference between the proportion of freshman who opposes plan and
the proportion of seniors who oppose it.**

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