Question

A political scientist obtained recordings of election-night acceptance speeches of seven newly elected representatives to the U.S. Congress. She counted the number of minutes devoted to urban problems in these speeches. Four of these representatives were from rural districts. They devoted 5, 0, 3, and 4 minutes to urban problems. The other three representatives studied, who were from urban districts, devoted 11, 11, and 14 minutes to urban problems. Do these results suggest that the amount of time devoted to urban problems in acceptance speeches of newly elected representatives to the U.S. Congress differ according to whether they come from rural or urban districts? (Use the .05 level.)

What is the mean of the distribution of differences between means?

1) What is the mean of the distribution of differences between means?

2) In the question above, what is t?

Answer 1

Given that,
mean(x)=3
standard deviation , s.d1=2.1602
number(n1)=4
y(mean)=12
standard deviation, s.d2 =1.7321
number(n2)=4
null, Ho: u1 = u2
alternate, H1: u1 != u2
level of significance, α = 0.05
from standard normal table, two tailed t α/2 =3.182
since our test is two-tailed
reject Ho, if to < -3.182 OR if to > 3.182
we use test statistic (t) = (x-y)/sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =3-12/sqrt((4.66646/4)+(3.00017/4))
to =-6.5008
| to | =6.5008
critical value
the value of |t α| with min (n1-1, n2-1) i.e 3 d.f is 3.182
we got |to| = 6.50085 & | t α | = 3.182
make decision
hence value of | to | > | t α| and here we reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -6.5008 ) = 0.007
hence value of p0.05 > 0.007,here we reject Ho
ANSWERS
---------------
1.
the mean of the distribution of differences between mean = 3-12 =-9
2.
null, Ho: u1 = u2
alternate, H1: u1 != u2
test statistic: -6.5008
critical value: -3.182 , 3.182
decision: reject Ho
p-value: 0.007
we have enough evidence to support the claim that difference in means between the amount of time devoted to urban problems in acceptance speeches of newly elected representatives to
the U.S. Congress differ according to whether they come from rural or urban districts.