Hypothesis Test Practice - CPP

[Pages:6]Hypothesis Test Practice 1) Eleven percent of the products produced by an industrial process over the past several months have failed to conform to specifications. The company modifies the process in an attempt to reduce the rate of nonconformities. In a random sample of 300 items from a trial run, the modified process produces 16 nonconforming items. Do these results provide convincing evidence that the modification of the process has been effective?

2) The germination rate of seeds is defined as the proportion of seeds that, when properly planted and watered, sprout and grow. A certain variety of grass seed usually has a germination rate of 0.80, and a company wants to see if spraying the seeds with a chemical that is known to change germination rates in other species will change the germination rate of this grass species. The company researchers spray 400 seeds with the chemical, and 307 of the seeds germinate. Does this data provide enough evidence that the chemical has changed the germination rate?

3) Nationally, the proportion of red cars on the road is 0.12. A statistically-minded fan of the Philadelphia Phillies (whose team color is red) wonders if fans who park at Citizens Bank Park (the Phillies home field) are more likely to drive red cars. One day during a home game, he takes an SRS of 210 cars parked in the lot while a game is being played, and counts 35 red cars. (There are 21,000 parking spaces.) Is this convincing evidence that Phillies fans prefer red cars more than the general population? Support your conclusion with a test of significance.

4) Do political "attack ads" work? A congressional candidate who currently has the support of only 44% of the voters runs a television spot that aggressively attacks the character of his opponent. He wants to know if the television spot has changed his support level among voters. The pollsters survey an SRS of 450 voters and find that 186 support the candidate. This produces a 95% confidence interval for the proportion of voters who support him of (0.368, 0.459). Is this convincing evidence that the attack ad has changed the support for this candidate?

5) LeRoy, a starting player for a major college basketball team, made only 40% of his free throws last season. During the summer, he worked on developing a softer shot in hopes of improving his free throw accuracy. In the first eight games of this season, LeRoy made 25 free throws in 40 attempts. You want to investigate whether LeRoy's work over the summer will result in a higher proportion of free-throw successes this season. What conclusion would you draw at the = 0.01 level about LeRoy's free throw shooting? Justify your answer with a complete significance test.

6) The Environmental Protection Agency has determined that safe drinking water should contain no more than 1.3 mg/liter of copper. You are testing water from a new source, and take 30 water samples. The mean copper content in your samples is 1.36 mg/l and the standard deviation is 0.18 mg/l. There do not appear to be any outliers in your data. Do these samples provide convincing evidence at the = 0.05 level that the water from this source contains unsafe levels of copper? Would your conclusion change if we instead used = 0.01?

7) Sweet corn of a certain variety is known to produce individual ears of corn with a mean weight of 8 ounces. A farmer is testing a new fertilizer designed to produce larger ears of corn, as measured by their weight. He finds that 32 randomly-selected ears of corn grown with this fertilizer have a mean weight of 8.23 ounces and a standard deviation of 0.8 ounces. There are no outliers in the data. Do these samples provide convincing evidence at the = 0.05 level that the fertilizer had a positive impact on the weight of the corn ears? Justify your answer.

8) The label on bottles of a certain brand of grapefruit juice say that they contain 180 ml of liquid. Your friend Jerry suspects that the true mean is less than that, so he takes an SRS of 40 bottles, and finds that the mean contents of his sample is 178.8 ml with standard deviation 5.4 ml. Does this provide convincing evidence that the mean contents of all bottles of this brand of grapefruit juice is less than 180 ml?

Solutions

1) Eleven percent of the products produced by an industrial process over the past several months have

failed to conform to specifications. The company modifies the process in an attempt to reduce the rate

of nonconformities. In a random sample of 300 items from a trial run, the modified process produces 16

nonconforming items. Do these results provide convincing evidence that the modification of the process

has been effective?

a)

H0: p = 0.11

Ha: p < 0.11

b)

= 16 = 0.05333

300

=

=

0.11,

=

0.11?0.89

300

=

0.01806

0.05333 - 0.11 = 0.01806 = -3.14 c)

P(z < -3.14) = 0.0008

d)

Since the p-value is so small (0.0008 < 0.05) we reject H0, there is strong evidence that the

proportion of nonconforming parts is now less than 0.11.

2) The germination rate of seeds is defined as the proportion of seeds that, when properly planted and

watered, sprout and grow. A certain variety of grass seed usually has a germination rate of 0.80, and a

company wants to see if spraying the seeds with a chemical that is known to change germination rates

in other species will change the germination rate of this grass species. The company researchers spray

400 seeds with the chemical, and 307 of the seeds germinate. Does this data provide enough evidence

that the chemical has changed the germination rate?

a)

H0: p = 0.80

Ha: p 0.80

b)

= 307 = 0.7675

400

=

=

0.80,

=

0.80?0.20

400

=

0.02

0.7675 - 0.80

=

0.02

= -1.625

c)

P(z < -1.63) = 0.0516, p-value = 0.0516x2 = 0.1032

d)

Since the p-value is so large (0.1032 > 0.05) we fail to reject H0, there is not enough evidence to

say that the germination rate is different from 0.80 after spraying with the chemical.

3) Nationally, the proportion of red cars on the road is 0.12. A statistically-minded fan of the

Philadelphia Phillies (whose team color is red) wonders if fans who park at Citizens Bank Park (the

Phillies home field) are more likely to drive red cars. One day during a home game, he takes an SRS of

210 cars parked in the lot while a game is being played, and counts 35 red cars. (There are 21,000

parking spaces.) Is this convincing evidence that Phillies fans prefer red cars more than the general

population? Support your conclusion with a test of significance.

a)

H0: p = 0.12

Ha: p > 0.12

b)

=

35 210

=

0.16667

=

=

0.12,

=

0.12?0.88

210

=

0.02242

0.16667 - 0.12 = 0.02242 = 2.08 c)

P(z > 2.08) = 1 - 0.9812 = 0.0188

d)

Since the p-value is so small (0.0188 < 0.05) we reject H0, we have strong evidence that the

proportion of red cars in the entire Phillies parking lot is greater than 0.12.

4) Do political "attack ads" work? A congressional candidate who currently has the support of only 44%

of the voters runs a television spot that aggressively attacks the character of his opponent. He wants to

know if the television spot has changed his support level among voters. The pollsters survey an SRS of

450 voters and find that 186 support the candidate. This produces a 95% confidence interval for the

proportion of voters who support him of (0.368, 0.459). Is this convincing evidence that the attack ad

has changed the support for this candidate?

a)

H0: p = 0.44

Ha: p 0.44

b) = 186 = 0.41333

450

=

=

0.44,

=

0.44?0.56

450

=

0.0234

0.41333 - 0.44 = 0.0234 = -1.14

c)

P(z < -1.14) = 0.1271, p-value = 0.1271x2 = 0.2542

d)

Since the p-value is so large (0.2542 > 0.05) we fail to reject H0, there is not enough evidence to

say that the attack ad has made support for this candidate different than 44%.

NOTE: Sorry about throwing the CI in this problem. The connection is that since 0.44 IS IN the

confidence interval, you fail to reject H0.

5) LeRoy, a starting player for a major college basketball team, made only 40% of his free throws last

season. During the summer, he worked on developing a softer shot in hopes of improving his free throw

accuracy. In the first eight games of this season, LeRoy made 25 free throws in 40 attempts. You want to

investigate whether LeRoy's work over the summer will result in a higher proportion of free-throw

successes this season. What conclusion would you draw at the = 0.01 level about LeRoy's free throw

shooting? Justify your answer with a complete significance test.

a)

H0: p = 0.40

Ha: p > 0.40

b)

=

25 40

=

0.625

=

=

0.40,

=

0.40?0.60

40

=

0.07746

0.625 - 0.40 = 0.07746 = 2.90 c)

P(z > 2.90) = 1 - 0.9981 = 0.0019

d)

Since the p-value is so small (0.0019 < 0.05) we reject H0, there is strong evidence that the

proportion of all free throws made after the summer training is more than 44%

6) The Environmental Protection Agency has determined that safe drinking water should contain no

more than 1.3 mg/liter of copper. You are testing water from a new source, and take 30 water samples.

The mean copper content in your samples is 1.36 mg/l and the standard deviation is 0.18 mg/l. There do

not appear to be any outliers in your data. Do these samples provide convincing evidence at the = 0.05

level that the water from this source contains unsafe levels of copper? Would your conclusion change if

we instead used = 0.01?

a)

H0: ? = 1.3

Ha: ? > 1.3

b)

= 1.36, n = 30, df = 29

=

=

1.3,

=

=

0.18 30

=

0.03286

1.36 - 1.3 = 0.03286 = 1.83

c)

P(t > 1.83) = 0.0388 by calculator, p-value between 0.025 and 0.05 by table

d)

at = 0.05: Since the p-value is so small (< 0.05) we reject H0, there is strong evidence that the

mean copper content in this water source is greater than 1.3 mg/liter

at = 0.01: Since the p-value is so large (> 0.01) we fail to reject H0, there is no enough evidence

to say that the mean copper content in this water source is greater than 1.3 mg/liter

7) Sweet corn of a certain variety is known to produce individual ears of corn with a mean weight of 8

ounces. A farmer is testing a new fertilizer designed to produce larger ears of corn, as measured by their

weight. He finds that 32 randomly-selected ears of corn grown with this fertilizer have a mean weight of

8.23 ounces and a standard deviation of 0.8 ounces. There are no outliers in the data. Do these samples

provide convincing evidence at the = 0.05 level that the fertilizer had a positive impact on the weight

of the corn ears? Justify your answer.

a)

H0: ? = 8

Ha: ? > 8

b)

= 8.23, n = 32, df = 31

=

=

8,

=

=

0.8 32

=

0.14142

8.23 - 8 = 0.14142 = 1.626

c)

P(t > 1.626) = 0.057 by calculator, p-value between 0.05 and 0.10 by table

d)

Since the p-value is so large (> 0.05) we fail to reject H0, there is not enough evidence that the

average weight for ears of corn using this new fertilizer is greater than 8 ounces

8) The label on bottles of a certain brand of grapefruit juice say that they contain 180 ml of liquid. Your

friend Jerry suspects that the true mean is less than that, so he takes an SRS of 40 bottles, and finds that

the mean contents of his sample is 178.8 ml with standard deviation 5.4 ml. Does this provide

convincing evidence that the mean contents of all bottles of this brand of grapefruit juice is less than

180 ml?

a)

H0: ? = 180

Ha: ? < 180

b)

= 178.8, n = 40, df = 39

=

=

180,

=

=

5.4 40

=

0.8538

178.8 - 180

= 0.8538 = -1.405

c)

P(t < -1.405) = 0.084 by calculator, p-value between 0.05 and 0.10 by table

d)

Since the p-value is so large (> 0.05) we fail to reject H0, there is not enough evidence that the

average content in all bottles of this brand of grapefruit juice is less than 180 ml

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