Hypothesis Testing

EQ: When is a result statistically significant?

Tests for Proportions

Test Statistic

z

For 1 sample proportions



p

( 1  

p p

)

n

Conditions

1. SRS 2. Population > 10n 3. np and n(1-p) > 10 For confidence intervals use p-hat, not p!

To find the p-value use normcdf.

In the book, Life in America’s Small Cities, GS Thomas reports that in 1990, 22.1% of all 16-19 year olds in Key West, Florida were high school dropouts. In 1995, a random sample of 193 people in this Key West age group showed 32 were dropouts. Does this indicate the proportion of high school dropouts in Key West is less than 22.1% Test at 5% significance.

1. Set up Hypotheses:

H o

:

p

 .

221

H A

:

p

 .

221

2.Test Statistic

1 proportion z-test Conditions: SRS Pop > 1,930 193(.221) = 42.7

193(.779) =150.3

 10

z

 

p

( 1 

p p

)

n

 .

166  .

221 (.

221 )(.

779 ) 193   1 .

842

3. Make a decision to reject/fail to reject based on p-values

P

(

z

  1 .

842 )  .

0327 Since .0327 < .05, Reject the null

4. Interpret

There is not enough evidence to find that the percent of dropouts in Key West is 22.1%. The percent appears to be lower.

In Computer Studies of the Humanities and Verbal Behavior, D Wishart and SV Leach found that 21.4% of passages in Plato’s Dialogues follow a particular syllable pattern. An owner of an antiquities store in Athens claims to have a manuscript that is a part of Dialogues. A random sample of 493 passages from the manuscript showed that 136 exhibited Plato’s syllable pattern. Do the data indicate this manuscript is part of Plato’s Dialogues?

1. Hypotheses:

H o

:

p

 .

214

H A

:

p

 .

214

2.Test Statistic

1 proportion z-test Conditions: SRS Pop > 4,930 493(.214) = 106 493(.786) = 387  10

z

 

p

( 1 

p p

)

n

 .

276  .

214 (.

214 )(.

786 ) 493  3 .

357

3. Make a decision to reject/fail to reject based on p-values

P

(

z

 3 .

357 or

z

 3 .

357 )  .

00078 Since .00078 < .05, Reject the null

4. Interpret

There is not enough evidence to find that the percent of passages with the pattern is 21.4%. It appears that the manuscript was not written by Plato

2005 AP FRQ

Some boxes of a certain brand of breakfast cereal include a voucher for a free video rental inside the box. The company that makes the cereal claims that a voucher can be found in 20 percent of the boxes. However, based on their experiences eating this cereal at home, a group of students believes that the proportion of boxes with vouchers is less than 0.2. This group of students purchased 65 boxes of the cereal to investigate the company’s claim. The students found a total of 11 vouchers for free video rentals in the 65 boxes.

Suppose it is reasonable to assume that the 65 boxes purchased by the students are a random sample of all boxes of this cereal. Based on this sample, is there support for the students’ belief that the proportion of boxes with vouchers is less than 0.2? Provided statistical evidence to support your answer.

1. Set up Hypotheses:

H o

:

p

 .

2

H A

:

p

 .

2

2.Test Statistic

1 proportion z-test Conditions: SRS Pop > 650 65(.169) = 11 65(.831) = 54  10

z

 

p

( 1 

p p

)

n

 .

169  .

2 (.

2 )(.

8 ) 65   .

62

3. Make a decision to reject/fail to reject based on p-values

P

(

z

  1 .

359 )  .

2676 Since .2676 > .05, Fail to reject the null

4. Interpret

There is not enough evidence to find that the proportion of boxes containing the vouchers differs from 20%.