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Lecture

# UASTAT141Ch20_21.pdf

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University of Alberta

Statistics

STAT141

Paul Cartledge

Winter

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Ch. 20 - Hypotheses and Test Procedures
Def’n: A null hypothesis is a claim about a population parameter that is assumed to be
true until it is declared false.
An alternative hypothesisis a claim about a population parameter that will be true
if the null hypothesis is false.
In carrying out a test of0H vs.AH , the hypothesis0H is “rejected” in favour oA H only if
sample evidence strongly suggests that H0is false. If the sample does not contain such
evidence, H0 is “not rejected” or you “fail to reject” it.
NEVER “accept” H or0H …foA different reasons.
Ex20.1) H : µ = 2.8 H : µ ≠ 2.8
0 A
↑ ↑
pop’n characteristic hypothesized value or “claim”
Def’n: A two-tailed test has “rejection regions” in both tails.
A one-tailed test has a “rejection region” in one tail.
loweAr-ailted has the “rejection region” in the left tail.
uppAer-ailted has the “rejection region” in the right tail.
Ex20.2)
a) H 0 µ = 15 HA: µ = 15 Æ INCORRECT
b) H 0 µ = 123 HA: µ = 125 Æ INCORRECT
c) H 0 µ = 123 HA: µ < 123 Æ CORRECT
d) H 0 µ ≥ 123 HA: µ < 123 Æ CORRECT
e) H 0 p = 0.4 HA: p > 0.6 Æ INCORRECT
f) H 0 p = 1.5 HA: p > 1.5 Æ INCORRECT
g) H 0 p= 0.1 HA: p ≠ 0.1 Æ INCORRECT
Two-Tailed Test Lower-Tailed Test Upper-Tailed Test
orSign for H0 == or ≥ = ≤
Sign for H ≠ < >
A
“Rejection region” In both tails In the left tail In the right tail
Ex20.3)
Is the mean different than 0? H 0: µ = µ0 H A: µ ≠ µ0
Is the mean lower than µ0? H 0: µ ≥ 0 H A: µ < µ0
Is the mean lower or still the sµ ? Hhan : µ ≤ µ H : µ > µ
0 0 0 A 0
Is the mean higher than µ0? H 0 µ ≤ µ0 H A: µ > µ0
Def’n: A test statistic the function of the sample data on which a conclusion to reject or
fail to reject0H is based. For example, Z and t are test statistics.
The P-value is a measure of inconsistency between the hypothesized value for a
pop’n characteristic and the observed sample. Assuming H 0 is true, the P-value can be
defined as the probability of obtaining a test statistic value at least as inconsisten0 with H
as what actually resulted. Keep in mind that we want to be inconsistent with 0to reject
it. Thus, the smaller the P-value, the more likely we reject H0. The significance level (denoted by α) is a number such that we reject H i0 the P-
value is less than or equal to that number.
The “significance level approach”:
reject H0if p-value ≤ α
do not reject H 0f p-value > α
Common choices for α are 0.01, 0.05, and 0.1, depending on the nature of the test.
PROBLEMS:
a) If you’re comparing to α = 0.05, are the P-values 0.045 and 0.000 001 “different”?
b) If we use a “cut-off” like α = 0.05, does it make sense to conclude differently between
P-values of 0.049 and 0.051?
Solution: ALWAYS report your P-value! That way a reader may draw their own
conclusions. Moreover, use the “judgment approach” for rejection. Here, there’s a
tendency of avoiding “cut-off” points and going toward some “acceptable” guidelines:
0.01 > P-value > 0 Æ strong to convincing evidence against H 0
0.05 > P-value > 0.01 Æ moderate to strong evidence against H
0
0.10 > P-value > 0.05 Æ suggestive to moderate evidence against H , ye0 inconclusive
1 > P-value > 0.1 Æ weak evidence against H 0
Steps of a Significance Test:
1. Assumptions: Specify variable/parameter. What assumptions apply? Do they hold?
2. Hypotheses: State the null/alternative hypotheses. (Select α for the test.)
3. Test statistic: Use the appropriate formula for the gi

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