Transcript Hypothesis Tests for Population Means

Worksheet for Hypothesis
Tests for Means
Section 10-4
1. A machine is designed to fill jars with 16 ounces of coffee. A
consumer suspects that the machine is not filling the jars completely.
They sampled 12 jars shown below. Is there enough evidence to
 claim at = 0.10?
support the consumer’s
15
15.4
16.2
16.1
15.8
16.2
15.7
15.6
16
16.3
15.3
15.9
1. A machine is designed to fill jars with 16 ounces of coffee. A consumer suspects
that the machine is not filling the jars completely. They sampled 12 jars shown
below. Is there enough evidence to support the consumer’s claim at α = 0.10?
µ = Means amount of ounces of coffee in a jar.
Assumptions:
Ho :   16

H A :   16
t
x

n
15.79  16
t
0.4055
12
t  1.79
1. SRS
2. Approx. Normal (Pop)
3. Independent: 10(12)=120
P  val  P ( x  15.79)  P (t  1.79)
tcdf ( , 1.79,11)
Pval  0.0505
Use a T-Test since σ is
unknown and n<30.
df=11
Reject the Ho since P-value (0.0505)<α (0.10).
There is sufficient evidence to support the claim that
the mean number of ounces of coffee is less than 16
ounces.
2. A researcher reports that the average salary of assistant professors
is more than $42,000. A sample of 30 assistant professors has a mean
salary of $43,260. At  = 0.05, test the claim that assistant professors
earn more than $42,000 a year. The standard deviation is $5,230.
.
2. A researcher reports that the average salary of assistant professors is more
than $42,000. A sample of 32 assistant professors has a mean salary of $43,260.
At α = 0.05, test the claim that assistant professors earn more than $42,000 a
year. The standard deviation is $5,230.
µ = Mean salary of assistant professors
Ho :   42000

H A :   42000
Assumptions:
1. SRS
2. Approx. Normal since n>30
3. Independent: 10(32)=320
z
x

n
43260  42000
z
5230
32
z  1.36
P  val  P ( x  43260)  P ( z  1.36)
ncdf (1.36, )
Pval  0.0869
Use a Z-Test since σ
is known
Fail toReject the Ho since P-value (0.087)>α (0.05).
There is insufficient evidence to support the claim that
the mean salary of assistant professors is more than
$42000.
3. The Medical Rehabilitation Foundation reports that the average cost of
rehabilitation for stroke victims is $24,672. to see if the average cost of
rehabilitation is different at a large hospital, a researcher selected a random
sample of 35 stroke victims and found that the average cost of their
rehabilitation if $25,266. The st. dev. Is $3,251. At
= 0.01, can it be
concluded that the average cost at a large hospital is different?

The Medical Rehabilitation Foundation reports that the average cost of rehabilitation
for stroke victims is $24,672. to see if the average cost of rehabilitation is different at
a large hospital, a researcher selected a random sample of 25 stroke victims and
found that the average cost of their rehabilitation if $25,266. The st. dev. Is $3,251.
At α = 0.01
µ = average cost of rehabilitation for stroke victims
Ho :   24672

H A :   24672
Assumptions:
1. SRS
2. Approx. Normal (Pop)
3. Independent: 10(25)=250
t
x

n
25266  24672
t
3251
25
t  0.91
P  val  P ( x  25266) * 2  P (t  0.91) * 2
tcdf (0.91, ,24) * 2
Pval  0.37
use T-test since
σ is unknown &
n< 30. df=24
Fail to Reject the Ho since P-value (0.37)>α (0.01).
There is insufficient evidence to support the claim that
the mean cost of rehabilitation for stroke victims is
different from $24,672.
4. A researcher wishes to test the claim that the average age of
lifeguards is Ocean City is greater than 24 years. She selects a sample
of 36 guards and finds the mean of the sample to be 24.7, with a st.
dev. Of 2 years. Is there evidence to support theclaim at
= 0.05?
4. A researcher wishes to test the claim that the average age of lifeguards is Ocean
City is greater than 24 years. She selects a sample of 26 guards and finds the mean
of the sample to be 24.7, with a st. dev. Of 2 years. Is there evidence to support the
claim at α = 0.05?
µ = Mean age of lifeguards in Ocean City.
Ho :   24

HA :   24
Assumptions:
1. SRS
2. Approx. Normal (pop)
3. Independent: 10(26)=260
t
x

n
24.7  24
t
2
26
t  1.78
P  val  P ( x  24.7)  P (t  1.78)
tcdf (1.78, ,25)
Pval  0.044
Use a t dist. Since σ is
unknown & n<30. df=25
Reject the Ho since P-value (0.044)<α (0.05).
There is sufficient evidence to support the claim that
the mean age of lifeguards is more than 24 years old.
5. The proportion of college students who gain weight their
first year is at least 65%. To test this, researchers sampled
200 students and found 130 had gained weight their first

year. Use a 5% significance level.
5. The proportion of college students who gain weight their first year is at least
65%. To test this, researchers sampled 200 students and found 120 had gained
weight their first year. Use a 5% significance level.
p = Pop. Prop. of college students who gain weight their first year.
Assumptions:
Ho : p  0.65

H A : p  0.65
1. SRS
2. Approx. Normal
Use a Z-Test for Prop.
z
z
pp
pq
n
0.6  0.65
0.65(0.35)
200
z  1.48
P  val  P ( p  0.6)  P ( z  1.48)
ncdf ( , 1.48)
Pval  0.069
np  200(.65)  130  5

nq  200(.35)  70  5
3. Independent
10(200)=2000
Fail to Reject the Ho since P-value (0069)>α (0.05).
There is insufficient evidence to support the claim that
the proportion of college students who gain weight
their first year is less than 65%.
6. A physician claims that jogger’s maximal volume oxygen uptake is
greater than the average of all adults. A sample of 15 joggers has a
mean of 43.6 ml per kg and a standard deviation of 6 ml/kg. If the
average of all adults is 36.7 ml/kg, is there enough evidence to support
the physician’s claim at  = 0.01?
6. A physician claims that jogger’s maximal volume oxygen uptake is greater than
the average of all adults. A sample of 15 joggers has a mean of 43.6 ml per kg and
a standard deviation of 6 ml/kg. If the average of all adults is 36.7 ml/kg, is there
enough evidence to support the physician’s claim at
= 0.01?
µ = Mean volume of oxygen uptake for joggers
Assumptions:
Ho :   36.7

H A :   36.7
1. SRS
2. Approx. Normal (Pop)
3. Independent: 10(15)=150
x
t
s
n
43.6  36.7
t
6
15
t  4.45
use t-test since σ is unknown & n<30.
P  val  P ( x  43.6)  P (t  4.45)
tcdf (4.45, ,14)
Pval  2.7 x10 4  0
Reject the Ho since P-value (0)<α (0.01).
There is sufficient evidence to support the claim that
the mean volume of oxygen uptake for joggers is
greater than 36.7m.per kg.
Df=14