PhStat
HYPOTHESIS TESTING
The setup
P-VALUE
 REJECT
NULL HYPOTHESIS WHEN
𝑝 − 𝑣𝑎𝑙𝑢𝑒 < 𝛼 (𝑙𝑒𝑣𝑒𝑙 𝑜𝑓 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒)
 DO
NOT REJECT NULL HYPOTHESIS WHEN
𝑝 − 𝑣𝑎𝑙𝑢𝑒 ≥ 𝛼 (𝑙𝑒𝑣𝑒𝑙 𝑜𝑓 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒)
Example 1 (𝜇) case1 𝜎 known
A manufacturer of sports equipment has
developed a new synthetic fishing line that
he claims has a mean breaking strength of
8 kilograms with a standard deviation of 0.5
kilogram. Test the hypothesis that  = 8
kilograms against the alternative that   8
kilograms if a random sample of 15 lines is
tested and is found to have a mean
breaking strength of 7.8 kilograms. Use a
0.01 level significance.
Example 2 (𝜇)
case 1 𝜎 unknown but n>30
It is claimed that an automobile is driven on
the average less than 20,000 kilometers per
year. To test this claim, a random sample of
100 automobile owners is asked to keep a
record of the kilometers they travel. Would
you agree with this claim if the random
sample showed an average of 23,500
kilometers and standard deviation of 3,900
kilometers? Use 0.01 level of significance.
Example 3 (𝜋)
A builder claims that heat pumps are
installed in 70% of all homes being
constructed today in the city of Richmond.
Would you agree with this claim if a random
survey of new homes in this city shows that 8
out of 15 had heat pumps installed?
Example 4 (𝜎 2 )
A manufacturer of car batteries
claims that the life of his batteries
have a standard deviation equal to
0.9 year. If a random sample of 10 of
these batteries have a standard
deviation of 1.2 years, do you think
that  > 0.9 year? Use a 0.05 level of
significance.
Example 5 (1-2 ) case 2
The following data represent the running
times of films produced by two motion
picture companies:
TIME (in minutes)
Company I 103 94 110 87
Company II 97 82 123 92
98
175 88 118
At the 0.05 level of significance, is there
evidence that the average running time of
motion pictures produced by company I is
longer than those produced by company
II? Assume equal variances.
Example 6 (1-2 ) case 3
Assuming unequal variances in EXAMPLE 5
we have:
Example 7 (1-2 ) case 4
A taxi company is trying to decide whether the
use of radial tires instead of regular belted tires
improves fuel economy.
Twelve cars were
equipped radial tires and driven over a prescribed
test course. Without changing drivers, the same
cars were then equipped with regular belted tires
and driven once again over the test course. The
gasoline consumption, in kilometers per liter, was
recorded as follows:
Car
Radial Tires
Belted Tires
1
2
3
4
5
6 7
8
4.2 4.7 6.6 7.0 6.7 4.5 5.7 6.0
4.1 4.9 6.2 6.9 6.8 4.4 5.7 5.8
9 10
7.4 4.9
6.9 4.7
11 12
6.1 5.2
6.0 4.9
Example 8 (𝜋1 − 𝜋2 )
A cigarette manufacturing firm
distributes two brands of cigarettes. If
it is found that 56 of 200 smokers
prefer brand A and that 29 of 150
smokers prefer brand B, can we
conclude at the 0.06 level of
significance that brand A outsells
brand B?
EXAMPLE 9
𝜎1 2
𝜎2 2
To verify the assumptions in EXAMPLE 5 we test whether the
variances are equal or not:
NOTE
𝑓1−𝛼 𝑣1 , 𝑣2
1
=
𝑓𝛼 𝑣2 , 𝑣1
The f-distribution table provided in class only
provides values for 𝑓𝛼 𝑥, 𝑦 so the above formula
helps you obtain values for 𝑓1−𝛼 𝑥, 𝑦