Significance Tests for a Proportion

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Question 1

Question 1.

True or false: $\widehat{p} $ has an approximate Normal distribution with mean, p, and standard deviation, $\sqrt{ \frac{p(1-p)}{n} } $, when n is large.

False

True

Incorrect. This is true provided np and n(1 - p) are both bigger than 10.

Correct. This is true provided np and n(1 - p) are both bigger than 10.

Incorrect. Try again.

Question 2

Question 2.

What is the problem with the test statistic formula $z= \frac{ \widehat{p}- p_{o} }{ \sqrt{ \frac{p(1-p)}{n} } } $ for testing the value of the parameter p?

We don't know the value of $\widehat{p} $ from the sample.

We don't know the value of p—that's why we are testing it.

We don't know the value of $p_{o} $ from the null hypothesis.

Incorrect. The problem is that we don't know the value of p. That's why we are performing a test of significance. Note: We can get the value of

from the sample and the value of

from the null hypothesis.

$\widehat{p}$ $p_{o}$

Correct. The problem is that we don't know the value of p. That's why we are performing a test of significance. Note: We can get the value of

from the sample and the value of

from the null hypothesis.

$\widehat{p}$ $p_{o}$

Incorrect. Try again.

Question 3

225

Question 3.

What do we check to determine whether n is large enough for a good Normal approximation?

$n \widehat{p} \geq 15$ and $n(1- \widehat{p} ) \geq 15$

$n p_{o} \geq 10$ and $n(1- p_{o} ) \geq 10$

$n \geq 30$

Incorrect. Since we assume the null hypothesis to be true, we assume that

so we use

as the value of p whenever we need a value for p.

$p= p_{o}$ $p_{o}$

Correct. Since we assume the null hypothesis to be true, we assume that

so we use

as the value of p whenever we need a value for p.

$p= p_{o}$ $p_{o}$

Incorrect. Try again.

Question 4

291

Question 4.

How should data be collected for a test on H₀: $p= p_{o} $ to be valid?

With a convenience sample

With a multistage sample

With a stratified sample

With a simple random sample

Incorrect. Data should be collected with a simple random sample.

Correct. Data should be collected with a simple random sample.

Incorrect. Try again.

Questions 5-6

332

Question 5.

The research question is, "Is the proportion of first-year students at this university who have "being very well-off financially" as an important personal goal significantly different from the national value of 73%?" What is the response variable and is it categorical or quantitative?

Number of students who feel "being very well-off financially" is an important personal goal which is quantitative

Percent of first-year students who feel "being very well-off financially" is an important personal goal which is quantitative

Whether first-year student feels "being very well-off financially" is an important personal goal which is categorical

Incorrect. The response variable is a measure on the individual. Each student is asked whether they feel "being very well-off financially is an important personal goal," so that is the response variable and it is categorical.

Correct. The response variable is a measure on the individual. Each student is asked whether they feel "being very well-off financially is an important personal goal," so that is the response variable and it is categorical.

Incorrect. Try again.

Question 6.

The research question is, "Is the proportion of first-year students at this university who have "being very well-off financially" as an important personal goal significantly different from the national value of 73%?" What is the appropriate alternative hypothesis for answering this question?

H_a:p > 0.73

H_a:p = 0.73

H_a:p $\neq$ 0.73

H_a:p < 0.73

Incorrect. The research question basically asks, "Is the proportion . . . significantly different from . . .73%?" Thus, the alternative hypothesis is p

0.73.

$\neq$

Correct. The research question basically asks, "Is the proportion . . . significantly different from . . .73%?" Thus, the alternative hypothesis is p

0.73.

$\neq$

Incorrect. Try again.

Question 7

379

Question 7.

True or false: We use the symbol p in the hypotheses because the response variable is categorical and we want to test proportion.

True

False

Incorrect. Whenever the response variable is categorical, we do inference on proportion.

Correct. Whenever the response variable is categorical, we do inference on proportion.

Incorrect. Try again.

Question 8

407

Question 8.

What do we check to determine whether n is large enough for a good Normal approximation when computing P-value?

n = 200 > 30

$n p_{o}$ = 200(0.73) = 146 > 10 and $n(1- p_{o} )$ = 200(1-0.73) = 54 > 10

$n \widehat{p} $ = 200(0.66) = 132 > 15 and $n(1- \widehat{p} )$ = 200(1-0.66) = 78 > 15

Incorrect. Since we assume the null hypothesis to be true, we assume that

= 0.73 so we use

= 0.73 and check

and

to see if n is large enough.

$p= p_{o}$ $p_{o}$ $n p_{o} \geq 10$ $n(1- p_{o} ) \geq 10$

Correct. Since we assume the null hypothesis to be true, we assume that

= 0.73 so we use

= 0.73 and check

and

to see if n is large enough.

$p= p_{o}$ $p_{o}$ $n p_{o} \geq 10$ $n(1- p_{o} ) \geq 10$

Incorrect. Try again.

Question 9

562

Question 9.

With P-value = 0.0258 and $\alpha$ = 0.05, should we reject H₀?

Yes

Incorrect. Because P-value = 0.0258 is less than

= 0.05, we reject H₀.

$\alpha$

Correct. Because P-value = 0.0258 is less than

= 0.05, we reject H₀.

$\alpha$

Incorrect. Try again.

Question 10

577

Question 10.

Given that the hypotheses are H₀: p = 0.73 and H_a: p $\neq$ 0.73 and we rejected H₀, what should we conclude?

The proportion of students who feel "being very well-off financially is an important personal goal" is 0.73.

We have insufficient evidence to conclude that the proportion of students who feel "being very well-off financially is an important personal goal" differs from 0.73.

The proportion of students who feel "being very well-off financially is an important personal goal" is significantly different from 0.73.

Incorrect. Since we rejected H₀, we conclude that H_a is correct. Thus the proportion differs significantly from 0.73.

Correct. Since we rejected H₀, we conclude that H_a is correct. Thus the proportion differs significantly from 0.73.

Incorrect. Try again.

Question 12

762

Question 12.

How were the data collected for this study?

Random sample of 6,000 three- to ten-year-olds with autism in Brick Township

Random sample of 6,000 three- to ten-year-olds in Brick Township

Sample of all 6,000 three- to ten-year-old children in Brick Township

Incorrect. All 6,000 three- to ten-year-olds in Brick Township were studied.

Correct. All 6,000 three- to ten-year-olds in Brick Township were studied.

Incorrect. Try again.

Question 13

1066

Question 13.

What table is used to find the P-value for a test of significance on proportion?

t table

Standard Normal table

Incorrect. We use the standard Normal table to find the P-value for testing proportion.

Correct. We use the standard Normal table to find the P-value for testing proportion.

Incorrect. Try again.

StatTutor Lesson - Significance Tests for a Proportion

Question 1

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Question 2

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Question 3

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Question 4

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Questions 5-6

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Question 7

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Question 8

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Question 9

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Question 10

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Question 11

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Question 12

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