AP Algebra

8/12/2019 AP Algebra

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Chapter 20 Testing Hypotheses About Proportions 335

Chapter 20Testing Hypotheses About Proportions

Chapter 20 Solutions to Class Examples

1. See Class Example 1.

2. Hypotheses The null hypothesis is that the proportion of M&Ms that are orange is20%. The alternative hypothesis is that the proportion of M&Ms that are orange issomething other than 20%. In symbols:

H 0 : p = 0.20

H A : p 0.20

Model Independence Assumption: It is reasonable to think that M&M colors are mutuallyindependent.

Random Sampling Condition: The 122 M&Ms we have can be consideredrepresentative of all M&Ms.10% Condition: 122 M&Ms is certainly less than 10% of all M&Ms.Success/Failure Condition: np = 122(0.20) = 48.8 and nq = 122(0.80) = 97.6 , whichare both at least 10, so the sample is large enough.The conditions are satisfied, so I can use the Normal model to perform a oneproportion z-test. Since we are testing the hypothesis that the proportion of orangeM&Ms is different than 0.20, we will use a 2 tailed test.

Mechanics n = 122 , x = 21 , 21122 0.172 p = ,

0.172 0.2

(0.2)(0.8)122

0.77

p p z

pqn

z

z

=

=

=

2 ( 0.172)

2 ( 0.77)

0.44

P P p

P z

=

Plan: Okay to use the Normal model because the trials are independent (random sample of blackAmericans), these 801 black Americans are less than 10% of all black Americans, and

( )0 801 (0.28) 224.28 10np = = and ( )0 801 (0.72) 576.72 10nq = = .

We will do a one-proportion z-test.

Mechanics: ( )( )0 00

0.28 0.72( ) 0.0159

801 p q

SD pn

= = = ; sample proportion: 0.38 p =

( )0.38 0.28( 0.38) ( ) 6.29 00.0159

P p P z P z

> = > = >

With a P-value so small (just about zero), I reject the null hypothesis that the proportion of blackAmericans who reported that there had been times in the last year when they had not been able toafford medical care is 0.28. There is enough evidence to suggest that the proportion of black

Americans who were not able to afford medical care in the past year is more than 28%.

2. Was your test one-tail upper tail, one-tail lower tail, or two-tail? Explain why youchose that kind of test in this situation.

One-tail, upper tail test. We are concerned that more than 28% of black Americans reported that therehad been times in the last year when they had not been able to afford medical care.

3. Explain what your P-value means in this context.

If the proportion of black Americans was not more than 28%, we could expect to find at least 38% ofthe 801 black Americans responding yes about 0% of the time (just about never).

Copyright 2012 Pearson Education, Inc. Publishing as Addison-Wesley


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338 Chapter 20 Testing Hypotheses About Proportions

Intro Stats Quiz B Chapter 20 Name

The International Olympic Committee states that the female participation in the 2000Summer Olympic Games was 42%, even with new sports such as weight lifting, hammerthrow, and modern pentathlon being added to the Games. Broadcasting and clothing

companies want to change their advertising and marketing strategies if the femaleparticipation increases at the next games. An independent sports expert arranged for arandom sample of pre-Olympic exhibitions. The sports expert reported that 202 of 454athletes in the random sample were women. Is this strong evidence that the participationrate may increase?

1. Test an appropriate hypothesis and state your conclusion.

2. Was your test one-tail upper tail, lower tail, or two-tail? Explain why you choose thatkind of test in this situation.




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Intro Stats Quiz B Chapter 20 Key

The International Olympic Committee states that the female participation in the 2000Summer Olympic Games was 42%, even with new sports such as weight lifting, hammerthrow, and modern pentathlon being added to the Games. Broadcasting and clothing

companies want to change their advertising and marketing strategies if the femaleparticipation increases at the next games. An independent sports expert arranged for arandom sample of pre-Olympic exhibitions. The sports expert reported that 202 of 454athletes in the random sample were women. Is this strong evidence that the participationrate may increase?


Hypothesis: 0 : 0.42 H p = : 0.42 A H p >

Plan: Okay to use the Normal model because the trials are independent (random sample of pre-Olympic exhibitions), these 454 athletes are less than 10% of all athletes at exhibitions, and

( )454 (0.42) 190.68 10np = = and ( )454 (0.58) 263.32 10nq = = . Use a N(0.42, 0.023) model,do a 1-proportion z-test.

Mechanics: ( )( )0 00

0.42 .58( ) 0.023

454 p q

SD pn

= = = ; sample proportion: 202 0.445454

p = =

0.445 0.42( 0.445) ( )

0.023P p P z

> = < =

( 1.09) 0.138P z < =

With a p-value (0.138) so large, I fail to reject thenull hypothesis that the proportion of femaleathletes is 0.42. There is not enough evidence tosuggest that the proportion of female athletes isincreasing.

2. Was your test one-tail upper tail, lower tail, or two-tail? Explain why you choose thatkind of test in this situation.

One-tail, upper test. The companies will change strategies only if there is strong evidence of anincrease in female participation rate from current rate of 42%.


If the proportion of female athletes has not increased, we could expect to find at least 202 females outof 454 pre-Olympic athletes about 13.8% of the time.



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340 Chapter 20 Testing Hypotheses About Proportions

Intro Stats Quiz C Chapter 20 Name

A company claims to have invented a hand-held sensor that can detect the presence ofexplosives inside a closed container. Law enforcement and security agencies are veryinterested in purchasing several of the devices if they are shown to perform effectively.

An independent laboratory arranged a test. They placed four empty boxes in the cornersof an otherwise empty room. For each trial they put a small quantity of an explosive inone of the boxes selected at random. The companys technician then entered the roomand used the sensor to try to determine which of the four boxes contained the explosive.The experiment consisted of 50 trials, and the technician was successful in finding theexplosive 16 times. Does this indicate that the device is effective in sensing the presenceof explosives?


2. Was your test one-tail upper tail, lower tail, or two-tail? Explain why you chose thatkind of test in this situation.




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Intro Stats Quiz C Chapter 20 Key

A company claims to have invented a hand-held sensor that can detect the presence ofexplosives inside a closed container. Law enforcement and security agencies are veryinterested in purchasing several of the devices if they are shown to perform effectively.

An independent laboratory arranged a test. They placed four empty boxes in the cornersof an otherwise empty room. For each trial they put a small quantity of an explosive inone of the boxes selected at random. The companys technician then entered the roomand used the sensor to try to determine which of the four boxes contained the explosive.The experiment consisted of 50 trials, and the technician was successful in finding theexplosive 16 times. Does this indicate that the device is effective in sensing the presenceof explosives?


Hypotheses: H 0 : p = 0.25 H a : p > 0.25

Plan: OK to use a Normal model because trials are independent (box is randomly chosen each time),and np = 12.5, nq = 37.5. Do a 1-proportion z-testMechanics: z = 1.14, P = 0.13Conclusion: With a P-value so high I fail to reject the null hypothesis. This test does not provideconvincing evidence that the sensor can detect the presence of explosives inside a box.

2. Was your test one-tail upper tail, lower tail, or two-tail? Explain why you chose thatkind of test in this situation.

One-tail, upper tail. The device is effective only if it can detect explosives at a rate higher than chance(25%).


Even if the device actually performs no better than guessing, we could expect to find the explosives 16or more times out of 50 about 13% of the time.



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