Setting:
Let x
1
, . . . , x n be a random sample from a population with mean µ
Questions:
What are plausible values of µ based on the data? Is the value µ = µ
0
(i.e., µ
0
= 7) consistent with the data?
1. Construct an α (100)% Confidence Interval for µ
¯ − z r s 2 n
, ¯ + z r s 2 n
!
where z = Φ(
1+ α
2
)
Interpretation: If we repeat the study 100 times and compute 100 α (100)% confidence intervals, we expect about
α (100) of the confidence intervals to contain the true, but unknown, mean µ .
2. Test the hypothesis: H
0
: µ = µ
0 vs.
H a
: µ = µ
0
.
Test Statistic: z =
¯ − µ
0 q s
2 n
Two-sided p-value:
2(1 − Φ( | z | ))
1

Interpretation : The p-value is the probability of observing an absolute difference between the sample average and
µ
0 at least as large as what we observed, if the null hypothesis that µ = µ
0 is true. If the p-value is small (i.e. smaller than 1 − α ), then we observed an unlikely outcome if the null hypothesis is true. When the p-value is small enough, we reject the null hypothesis in favor of the alternative that
µ = µ
0
.
Connection: The p-value from the two-sided test of H
0 is bigger than 1 − α if and only if µ
0 is contained in the
:
α
µ = µ
0
(100)%
CI for µ . The confidence interval contains the values of µ that are consistent with the data. We fail to reject the null hypothesis that µ = µ
0 when a mean of µ
0 is plausible, based on the data.
Setting:
Let x
1
, . . . , x n x with mean µ
1 be a random sample of size and let y
1
, . . . , y n y with mean µ
2 be a random sample of size n n x y from a population from a population
Questions: What are plausible values for the difference µ
1
− µ
2
, based on the data? Is the difference µ
1
− µ
2
= d
0
(i.e., d
0
= 0) consistent with the data?
1. Construct an α (100)% confidence interval for the difference µ
1
−
µ
2
.


¯ − y − z s s 2
1 n
1
+ s 2
2 n
2
, ¯ − y + z s s 2
1 n
1
+
 s 2
2 n
2
 where z = Φ(
1+ α
2
)
2

Interpretation: If we repeat the study 100 times and compute 100 confidence intervals in this way, we expect α (100) of the confidence intervals to contain the true, but unknown, value of µ .
2. Test the hypothesis H
0
: µ
1
− µ
2
= d
0 vs.
H
0
: µ
1
− µ
2
= d
0
.
Test Statistic: z =
¯ − ¯ − d
0 q s
2
1 n
1
+ s
2
2 n
2
Two-sided p-value:
2(1 − Φ( | z | ))
Interpretation: The p-value is the probability of observing an absolute difference between ¯ − y and d
0 that is at least as great as what we observed, if the null hypothesis that µ
1
− µ
2
= d
0 is true. If the p-value is small (i.e. smaller than 1 − α ), then we observed an unlikely outcome if the null hypothesis is true. When the p-value is small enough, we reject the null hypothesis in favor of the alternative that
µ
1
− µ
2
= d
0
.
Connection: The p-value from the two-sided test of H
0
: µ
1
−
µ
2
= d
0 is bigger than 1 − α if and only if d
0 is contained in the
α (100)% CI for µ
1
− µ
2
. The confidence interval contains the values of the difference between the two population means that are consistent with the data. We fail to reject the null hypothesis that µ
1
− µ
2
= d
0 when d
0 is a plausible value for the true, but unknown, mean difference, based on the data.
3