Homework

Worksheet on one Sample Proportion Hypothesis Tests

In a poll, 432 of 1004 adults surveyed said that they believe the Census Bureau when it says the information you give is kept confidential. Is there convincing evidence that fewer than half of U.S. adults believe the information is kept confidential. Use a 0.01 significance level.

#1

In a poll, 432 of 1004 adults surveyed said that they believe the Census Bureau when it says the information you give is kept confidential. Is there convincing evidence that fewer than half of U.S. adults believe the information is kept confidential. Use a 0.01 significance level.

p = Pop. Prop. of adults who believe the Census information is kept confidential.

: 0.50

: 0.50o

A

H p

H p

Assumptions:

1. SRS

2. Approx. Norma

3. 10(1004)=10040 (Independent

1004(.5) 502 5

1004(.5) 502 5

np

nq

0.43 0.50

0.5(0.5)1004

4.44

p pz

pqn

z

z

6

( 0.43) ( 4.44)

( , 4.44)

4.5 10 0

P val P p P z

ncdf

Pval x

Reject the Ho since P-value (0)<α (0.01).

There is sufficient evidence to support the claim that less than half of Americans believe that the Census information is kept confidential.

4.44

APL patients were given an arsenic compound as part of their treatment. Of those receiving the arsenic, 42% were in remission and showed no signs of leukemia in later exams. It is know that 15% of APL patients go into remission after conventional treatment. Suppose the study included 100 randomly selected patients. Is there sufficient evidence to conclude that the proportion in remission is greater than 0.15? Use a 0.01 significance level.

#2

.

p = Pop. Prop. of APL patients who go into remission.

: 0.15

: 0.15o

A

H p

H p

Assumptions:

1. SRS

2. Approx. Normal

3. 10(100) = 1000 Independent

100(.15) 15 5

100(.85) 85 5

np

nq

0.42 0.15

0.15(0.85)100

7.56

p pz

pqn

z

z

14

( 0.42) ( 7.56)

(7.56, )

2.03 10 0

P val P p P z

ncdf

Pval x

Reject the Ho since P-value (0)<α (0.01).

There is sufficient evidence to support the claim that the proportion in remission is greater than 15%.

Appoximately 42% of the 100 randomly selected APL patient who received arsenic were in remission. We know that 15% of APL patients go into remission after conventional treatment. Is there sufficient evidence to conclude that the proportion in remission is greater than 0.15? Use a 0.01.

7.56

The Associated Press reported that 715 of Americans, age 25 and older, are overweight, a substantial increase over the 58% figure from 1983. Although this information came from a Harris Poll rather than a census of the population, let’s assume for the purp0ses of this exercise that the nationwide population proportion is exactly 0.71. suppose that an investigator wishes to know whether the proportion of such individuals in her state who are overweight differs from the national proportion. A random sample of size n = 600 results in 450 who are classified as overweight.

a. What can the investigator conclude? Answer this question by carrying out a hypothesis

test with a 1% significance level.

b. Describe Type I and Type II error

c. Which type of error might you be making?

#3

The population proportion is exactly 0.71. An investigator want to check to see if her state’s proportion is different. Out of 600, 450 are overweight. Use 1% significance level.

p = Pop. Prop. of who are considered overweight.

: 0.71

: 0.71o

A

H p

H p

Assumptions:

1. SRS

2. Approx. Normal

3. 10(600)=6000 Independent

600(.71) 426 5

600(.29) 174 5

np

nq

0.75 0.71

0.71(0.29)600

2.16

p pz

pqn

z

z

( .75) * 2 ( 2.16) * 2

(2.16, ) * 2

0.030

P val P p P z

ncdf

Pval

Fail to Reject the Ho since P-value (0.03)>α (0.01).

There is insufficient evidence to support the claim that the proportion of overweight people is different from nationwide proportion of 71%

2.16

Georgia’s HOPE scholarship program guarantees fully paid tuition to Georgia public universities for Georgia high school seniors who have a B averrage in academic requirements as long as they maintain a B average in college. Of 137 randomly selected students enrolling in the Ivan Allen College at the Georgia Institute of Technology in 1996 who had a B average going into colllege, 53.2% had a GPA below 3.0 at the end of their first year. Do these data provide convincing evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship?

#4

.

p = Pop. Prop. of Ivan College students who lose their scholarship

: 0.5

: 0.5o

A

H p

H p

Assumptions:

1. SRS

2. Approx. Normal

3. 10(137)=1370 Independent

137(.5) 68.5 5

137(.5) 68.5 5

np

nq

0.532 0.5

0.5(0.5)137

0.75

p pz

pqn

z

z

( 0.532) ( 0.75)

(0.75, )

0.2266

P val P p P z

ncdf

Pval

Fail to Reject the Ho since P-value (0.2266)>α (0.05).

There is insufficient evidence to support the claim that the proportion of Ivan College students who lose their scholarship is more than 50%.

Of 137 randomly selected students at Ivan, 53.2% had a GPA below 3.0 at the end of their first year. Do these data provide convincing evidence that a majority of students at Ivan Allen College who enroll with a HOPE scholarship lose their scholarship?

0.75

Teenagers (age 15 to 20) make up 7% of the driving population. A study of auto accidents conducted by the Insurance Institute for Highway Safety found that 14% of the accidents studied involved teenage drivers. Suppose that this percentage was based on examining records from 500 randomly selected accidents. Does the study provide convincing evidence that the proportion of accidents involving teenage drivers is greater than 7%, the proportion of teenage drivers?

#5

.

p = Pop. Prop of auto accidents involving teenage drivers

Assumptions:

1. SRS

2. Approx. Norma

3. 10(500)=5000 - Independent

Reject the Ho since P-value (0)< α (0.05).

There is sufficient evidence to support the claim that the proportion of teenage drivers involved in accidents is greater than the proportion of teenage drivers (7%).

Sample of 500 had 14% of accidents had teenage drivers. Test at 5 % significance against the population proportion of teenage drivers which is 7%.

Students at the Akademia Podlaka conducted an experiment to determine whether the Belgium-minted Euro coin was equally likely to land heads up or tails up. Coins were spun on a smooth surface, and in 250 spins, 140 landed with the heads side up. Test this using a 1% significance level.

#6

.

p = Pop. Prop of auto accidents involving teenage drivers

Assumptions:

1. SRS

2. Approx. Norma

3. 10(500)=5000 - Independent

Fail to Reject the Ho since P-value (0.0574)> α (0.01).

There is insufficient evidence to support the claim that the coin is not equally likely to land on heads as tails.

140 out of 250 land on heads up. Is it equally likely. Test using a 1% significance level.