Question

Suppose you want to test the claim the paired sample data given below come from a population for which the mean difference is ??=0.

x 73 59 88 59 67 86 51

y 67 65 80 55 76 67 57

Use a 0.05 significance level to find the following:
(a)    The mean value of the difference ? for the paired sample data
d¯=
(b)    The standard deviation of the differences ? for the paired sample data
??=
(c)    The t-test statistic
?=
(d)    The positive critical value
?=
(e)    The negative critical value
?=
(f)    Does the test statistic fall in the critical region?

A. No
B. Yes

(g)    Construct a 95% confidence interval for the population mean of all differences x−y.
<??<

Answer 1

a)

dbar = 2.29

b)

sd = 9.9451

c)

Test statistic,
t = (dbar - 0)/(s(d)/sqrt(n))
t = (2.29 - 0)/(9.9451/sqrt(7))
t = 0.609

d)

The positive critical value = 2.447

e)

The negative critical value = -2.447

f)

No

g)

sample mean, xbar = 2.29
sample standard deviation, s = 9.9451
sample size, n = 7
degrees of freedom, df = n - 1 = 6

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.447

ME = tc * s/sqrt(n)
ME = 2.447 * 9.9451/sqrt(7)
ME = 9.198

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (2.29 - 2.447 * 9.9451/sqrt(7) , 2.29 + 2.447 * 9.9451/sqrt(7))
CI = (-6.91 , 11.49)

-6.91< mud < 11.49