worksheet for hypothesis tests for 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 support the consumer’s claim at = 0.10?

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.

: 16

: 16o

A

H

H

Assumptions:

1. SRS

2. Approx. Normal (Pop)

3. Independent: 10(12)=120

Use a T-Test since σ is unknown and n<30. df=11

15.79 160.4055

121.79

xt

n

t

t

( 15.79) ( 1.79)

t ( , 1.79,11)

0.0505

P val P x P t

cdf

Pval

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.

.

µ = Mean salary of assistant professors

: 42000

: 42000o

A

H

H

Assumptions:

1. SRS

2. Approx. Normal since n>30

3. Independent: 10(32)=320

Use a Z-Test since σ is known

43260 420005230

321.36

xz

n

z

z

( 43260) ( 1.36)

(1.36, )

0.0869

P val P x P z

ncdf

Pval

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.

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.

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

: 24672

: 24672o

A

H

H

Assumptions:

1. SRS

2. Approx. Normal (Pop)

3. Independent: 10(25)=250

25266 246723251

250.91

xt

n

t

t

( 25266) * 2 ( 0.91) * 2

t (0.91, ,24) * 2

0.37

P val P x P t

cdf

Pval

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.

use T-test since σ is unknown & n< 30. df=24

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.

: 24

: 24o

A

H

H

Assumptions:

1. SRS

2. Approx. Normal (pop)

3. Independent: 10(26)=260

Use a t dist. Since σ is unknown & n<30. df=25

24.7 242

261.78

xt

n

t

t

( 24.7) ( 1.78)

t (1.78, ,25)

0.044

P val P x P t

cdf

Pval

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.

p = Pop. Prop. of college students who gain weight their first year.

: 0.65

: 0.65o

A

H p

H p

Assumptions:

1. SRS

2. Approx. Normal

3. Independent 10(200)=2000

200(.65) 130 5

200(.35) 70 5

np

nq

Use a Z-Test for Prop.

0.6 0.65

0.65(0.35)200

1.48

p pz

pqn

z

z

( 0.6) ( 1.48)

( , 1.48)

0.069

P val P p P z

ncdf

Pval

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%.

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.

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

: 36.7

: 36.7o

A

H

H

Assumptions:

1. SRS

2. Approx. Normal (Pop)

3. Independent: 10(15)=150

use t-test since σ is unknown & n<30.

43.6 36.76

154.45

xt

sn

t

t

4

( 43.6) ( 4.45)

t (4.45, ,14)

2.7 10 0

P val P x P t

cdf

Pval x

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