Student: Instructor: Doug Ensley Assignment: Date: Course...

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Student: _____________________Date: _____________________

Instructor: Doug EnsleyCourse: Applied Statistics - Ensley 03/04 Assignment: Online K - Chi Squared

In the mid-1800s, a doctor decided to make the doctors wash their hands with a strong disinfectant between patients at a clinic with a death rate of %. The doctor wanted to test the hypothesis that the death rate would go down after the newhand-washing procedure was used. What null and alternative hypotheses should he have used? Explain, using both words and symbols. Explain the meaning of any symbols you use.

14.2

State the null and alternative hypotheses using words.

A. : The death rate has remained the same at % after starting hand-washing.H0 14.2: The death rate has increased to a value greater than %.HA 14.2

B. : The death rate has remained the same at % after starting hand-washing.H0 14.2: The death rate has decreased to a value less than %.HA 14.2

C. : The death rate has increased to a value greater than % after starting-hand washing. H0 14.2: The death rate has decreased to a value less than %.HA 14.2

D. : The death rate has remained the same at % after starting hand-washing.H0 14.2: The death rate has changed from %.HA 14.2

State the null and alternative hypotheses using symbols. Choose the correct answer below.

A. : p , : p (p is the proportion of deaths at the clinic.)H0 = 0.142 HA ≠ 0.142

B. : p , : p (p is the proportion of deaths at the clinic.)H0 > 0.142 HA < 0.142

C. : p , : p (p is the proportion of deaths at the clinic.)H0 = 0.142 HA < 0.142

D. : p , : p (p is the proportion of deaths at the clinic.)H0 = 0.142 HA > 0.142

(1) (2) (3) (4) (5)

An automobile association reported in that % of people in a certain region surveyed had admitted to driving impaired at least once in the past year. Suppose you are to survey randomly chosen people from this region to determine whether the proportion who have driven impaired at least once in the past year has from .

2009 34

increased 2009a. Give the null and alternative hypotheses to test whether the proportion of people from this region who have driven while impaired at least once in the past year has from .increased 2009b. The statistical hypothesis in part a is a _________ (right-tailed, left-tailed, or two-tailed) test.

a. Write the hypotheses in symbols.

: (1) (2) H0

: (3) (4) HA(Type integers or decimals. Do not round.)

b. The statistical hypothesis in part a is a (5) test.

pp0

p

=>≠<

p0

pp

≠=><

left-tailedright-tailedtwo-tailed

Online K - Chi Squared-Doug Ensley https://xlitemprod.pearsoncmg.com/api/v1/print/math

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*3.

(1) (2) (3) Do not rejectReject

(4) greater thanequal toless than

Conventional wisdom suggests that half of Canadians ( %) have supported the existence of the long-gun registry. The same wisdom suggests that the support for the long-gun registry is much lower in a certain large region. A poll of Canadians living in this region showed that support the long-gun registry. Assume that this survey was a random sample. Test the claim that the proportion of Canadians living in this region who are in favour of the long-gun registry is less than 50%. Perform a hypothesis test using a significance level of . Choose one of the following conclusions.

50250

64

0.05

i. The proportion is not significantly different from %. (A significant difference is one for which the p-value is less than or equal to .)

500.05

ii. The proportion is significantly less than %.50

Determine the null and alternative hypotheses. Let p be proportion in the region who support the existence of the long-gunregistry.

: p (1) H0

: p (2) HA(Type integers or decimals. Do not round.)

Determine the test statistic.

(Round to two decimal places as needed.)

Determine the p-value.

(Round to three decimal places as needed.)

Determine the result of the hypothesis test.

(3) because the p-value is (4) the significance level.H0

Choose the correct conclusion below.

A. The proportion %.is not significantly different from 50

B. The proportion %.is significantly less than 50

=≠><

<≠=>


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*4. Research subjects were followed from adolescence to adulthood, and all the people in the sample were categorized in terms of whether they were obese and whether they

Dating Cohabiting MarriedObese 85 104 150

Not Obese 358 325 278

were dating, cohabiting, or married. Test the hypothesis that relationship status and obesity arerelated, using a significance level of 0.05. Can we conclude from these data that living with someone is making some people obese and that marrying is making people even more obese?Can we conclude that obesity affects relationship status? Explain.

Choose the correct null hypothesis ( ) and alternative hypothesis ( ).H0 HA

A. : Obesity and relationship status have equal proportions.H0: Obesity and relationship status have non-equal proportions.HA

B. : Obesity and relationship status have a p-value of 0.05.H0

C. : Obesity and relationship status are independent.H0: Obesity and relationship status are related.HA

D. : Obesity and relationship status are related.H0: Obesity and relationship status are independent.HA

Calculate the chi-square test statistic.

(Round to two decimal places as needed.)

Determine the p-value.

(Round to three decimal places as needed.)

Reject or do not reject the null hypothesis, and state what that means.

A. and conclude that obesity and relationship status related.Fail to reject H0 are

B. and conclude that obesity and relationship status related.Reject H0 are not

C. and conclude that obesity and relationship status related.Fail to reject H0 are not

D. and conclude that obesity and relationship status related.Reject H0 are

Can we conclude from these data that living with someone is making some people obese and that marrying is making people even more obese? Can we conclude that obesity affects relationship status? Explain why or why not.

A. Yes. Since cohabiting and married people have a higher rate of obesity, it is clear that being obese increases the chances of getting married.

B. No. Although a relationship is evident, this is an observational study and cannot be used to prove causality.

C. No. Obesity rates actually decrease in cohabitating and married people.

D. Yes. Since the obesity rate is higher in cohabiting and married people, it can be concluded that marriage can cause obesity.


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*5.

6.

In a hypothesis test to determine if there is a relationship between two categorical variables, what can be said about the null hypothesis?

Choose the correct answer below.

A. The null hypothesis always contains an equals sign.

B. The null hypothesis changes depending on the context of the problem.

C. The null hypothesis always states that there is no relationship between the variables.

D. The null hypothesis always states that there is a relationship between the variables.

For the given 2 3 table on gender and happiness,

software tells us that and the P-value . Complete parts a and b below.

×

X2 = 2.264= 0.32

HappinessGender Not Pretty VeryFemale 177 596 311

Male 131 536 270

a. State the null hypothesis ( ) and alternative hypothesis ( ), in context, to which these results apply. Choose the correct answer below.

H0 Ha

A. : Gender and happiness are dependentH0: Gender and happiness are independentHa

B. : P(not happy) P(pretty happy) P(very happy)H0 = =: P(not happy) P(pretty happy) P(very happy)Ha ≠ ≠

C. : Gender and happiness are independentH0: Gender and happiness are dependentHa

D. : Gender and happiness have equal proportionsH0: Gender and happiness have non-equal proportionsHa

b. Interpret the P-value.

A. and conclude that gender and happiness independent.Fail to reject H0 could be

B. and conclude that gender and happiness independent.Fail to reject H0 are not

C. and conclude that gender and happiness independent.Reject H0 could be

D. and conclude that gender and happiness independent.Reject H0 are not


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7. According to a study, of males and of females indicated a belief in life after death. Complete parts (a)through (c).

891 1123 1272 1487

a. Construct a contingency table relating gender (categories male and female) as the rows to belief about life after death(categories yes and no) as the columns.

BELIEF IN LIFE AFTER DEATHGENDER Yes No Total

Males

Females

b. Find the four expected cell counts for the chi-squared test.

BELIEF IN LIFE AFTER DEATHGENDER Yes No Total

Males 1123

Females 1487

(Round to the nearest hundredth as needed.)

Compare the four expected cell counts to the observed cell counts, identifying cells having more observations than expected.

A. There are more men and women who do not believe in the afterlife than is expected.

B. There are more men and women who believe in the afterlife than is expected.

C. There are more women who believe in the afterlife and men who do not than is expected.

D. There are more men who believe in the afterlife and women who do not than is expected.

c. Find the X test statistic value to summarize how far the observed cell counts fall from the expected cell counts.2

The X test statistic value is . (Round to the nearest tenth as needed.)2


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8. The given table refers to a survey in which senior high school students were randomly sampled. It cross-tabulates whether a student had ever smoked cigarettes and whether a student had ever used marijuana. Analyze these data by (a) finding and interpreting conditional distributions with marijuana use as the response variable and (b) reporting all five steps of the chi-squared test of independence.

MarijuanaCigarettes Yes No

Yes 926 574No 41 716

a. Input each percentage in the table to the right.

(Round to one decimal place as needed.)

MarijuanaCigarettes Yes No

Yes % %

No % %

Choose the correct interpretation below.

A. The data suggest that cigarette use is much more common for those who have smoked marijuana than for those who have not.

B. The data suggest that cigarette use is no different for those who have smoked marijuana than for those who have not.

C. The data suggest that marijuana use is much more common for those who have smoked cigarettes than for those who have not.

D. The data suggest that marijuana use is no different for those who have smoked cigarettes than for those who have not.

b. Choose the correct assumptions that are made below.

A. There are four categorical variables, randomization was used to obtain the data, and the expected count was at least

B. There are four categorical variables, randomization was used to obtain the data, and the expected count was at least ten in all cells.

C. There are two categorical variables, randomization was used to obtain the data, and the expected count was at least five in all cells.

D. There are two categorical variables, randomization was used to obtain the data, and the expected count was at least ten in all cells.

Choose the correct null hypothesis ( ) and alternative hypothesis ( ) below.H0 Ha

A. : Cigarette use and marijuana use have a P-value 0.05.H0 =: Cigarette use and marijuana use have a P-value 0.05.Ha ≠

B. : Cigarette use and marijuana use are independent.H0: Cigarette use and marijuana use are dependent.Ha

C. : Cigarette use and marijuana use are dependent.H0: Cigarette use and marijuana use are independent.Ha

D. : Cigarette use and marijuana use have equal proportions.H0: Cigarette use and marijuana use have non-equal proportions.Ha

Use technology to find the chi-squared test statistic.

(Round to the nearest integer as needed.)X2 =

Use technology to find the P-value.

(Round to four decimal places as needed.)

Use the P-value found above and a significance level of 0.05 to form a conclusion. Choose the correct conclusion below.

A. and conclude that the two variables are dependent.Fail to reject H0


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9.

and conclude that the two variables independent.Reject H0 could be

1: Technology Output

Senior high school students in a large city were randomly sampled and surveyed about whether they had ever used cigarettes and whether they had ever consumed alcohol. The accompanying technology output shows the results. Complete parts a and b below.

Click the icon to view the technology output.1

a. True or false: Using cigarette use as the column variable and alcohol use as the row variable results in different values for the chi-squared statistic and the P-value shown in the output. Choose the correct answer below.

A. This is false. The steps of the test and the results are identical no matter which of the two categorical variables defines the rows or columns.

B. This is true. The new chi-squared statistic will be a value different from and unrelated to the current chi-squaredstatistic.

C. This is true. The new chi-squared statistic will be the reciprocal of the current chi-squared statistic.

D. This is true. The new chi-squared statistic will be the opposite of the current chi-squared statistic.

b. Explain what values would be obtained for the z-statistic and P-value when conducting a significance test of : p1 p2against : p1 p2, where p1 is the population proportion of non-cigarette users who have drunk alcohol and p2 is the population proportion of cigarette users who have drunk alcohol.

H0 =Ha ≠

The z-statistic would be .(Round to two decimal places as needed.)

The P-value would be .(Round to three decimal places as needed.)

Row: Cigarette Columns: Alcohol

No Yes No 271 510Yes 53 1446

Pearson Chi-Square = 409.038DF , P-Value= 1 = 0.000


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10. A study was done in which subjects who had suffered from a stroke were given either placebo or aspirin. Three years later,it was found that of the subjects taking placebo and of the subjects taking aspirin had a heart attack. Use the given information to complete parts a and b.

31 690 20 680

Click the icon to view the chi-squared distribution table.2

a. Report the data in the form of a 2 2 contingency table.×

Heart AttackTreatment Yes No Total

PlaceboAspirin

Total

b. Carry out all five steps of the null hypothesis that having a heart attack is independent of whether one takes placebo or aspirin.

Choose the correct assumptions that are made.

A. There are two categorical variables, randomization was used to obtain the data, and the expected count was at least five in all cells.

B. There are two categorical variables, randomization was used to obtain the data, and the expected count was at least ten in all cells.

C. There are four categorical variables, randomization was used to obtain the data, and the expected count was at least ten in all cells.

D. There are four categorical variables, randomization was used to obtain the data, and the expected count was at least five in all cells.

Choose the correct null hypothesis ( ) and alternative hypothesis ( ).H0 Ha

A. : Taking aspirin and having a heart attack have equal proportions.H0: Taking aspirin and having a heart attack have non-equal proportions.Ha

B. : Taking aspirin and having a heart attack are dependent.H0: Taking aspirin and having a heart attack are independent.Ha

C. : Taking aspirin and having a heart attack are independent.H0: Taking aspirin and having a heart attack are dependent.Ha

D. : Taking aspirin and having a heart attack have a P-value 0.05.H0 =: Taking aspirin and having a heart attack have a P-value 0.05.Ha ≠

Calculate the chi-squared test statistic.

(Round to three decimal places as needed.)χ2 =

Choose the correct P-value.

A. 0.2796

B. 0.0876

C. 0.0292

D. 0.1293

Use the P-value found above and a significance level of 0.05 to form a conclusion.

A. and conclude that the two variables .Reject H0 are dependent or associated

B. and conclude that the two variables .Fail to reject H0 are dependent or associated


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2: chi-squared distribution table


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11. In an experiment on chlorophyll inheritance in corn, for seedlings of self-fertilized heterozygous green plants,seedlings were green and were yellow. Theory predicts that 75% of the seedlings would be green. Complete parts (a)through (c).

1087 841246

a. Specify a null hypothesis for testing the theory. Choose the correct answer below.

A. versus H : p = 0.250 H : p ≠ 0.25a

B. versus H : p = 0.750 H : p > 0.75a

C. versus H : p = 0.750 H : p ≠ 0.75a

D. versus H : p = 0.750 H : p < 0.75a

b. Find the value of the chi-squared goodness-of-fit statistic.

(Round to the nearest tenth as needed.)X =2

Report the df value of this chi-squared statistic.

df =

c. Report the P-value.

P-value (Round to the nearest hundredth as needed.)=

Interpret the P-value.

A. The probability of obtaining a test statistic as extreme as that observed, assuming the null hypothesis is true, is . There is evidence against the null hypothesis, but not very strong.

0.07

B. The probability of obtaining a test statistic as extreme as that observed, assuming the alternative hypothesis is true, is . There is evidence against the null hypothesis, but not very strong.0.07

C. The probability of obtaining a test statistic as extreme as that observed, assuming the null hypothesis is true, is . There is very strong evidence against the alternative hypothesis.

0.07


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12.

3: Birthday Data

(1)

(2)

(3) Do not rejectReject

(4) is notis

Based on a random sample of students taken at a college, the accompanying table shows how many had their birthday in the first, second, third, or fourth quarter of the year. Is there evidence that the probabilities of having a birthday in a given quarter are not equal? Complete parts a through d below.

86

Click the icon to view the birthday data.3

a. Formulate the null and alternative hypotheses.

: (1) H0

: (2) Ha

b. Find the expected values and compute the chi-squared statistic.

Jan-Mar Apr-Jun Jul-Sep Oct-DecObserved 19 22 32 13Expected(Round to three decimal places as needed.)

X2 =(Round to three decimal places as needed.)

c. The test will be based on how many degrees of freedom?

df =

d. Find the P-value and write a conclusion in context.

P-value =(Round to three decimal places as needed.)

State the conclusion. Use a significance level of .0.05

(3) . There (4) sufficient evidence to conclude that (5) H0

Birthday Jan-Mar Apr-Jun Jul-Sep Oct-Dec n

19 22 32 13 86

P(Jul-Sep) is greater than the other probabilities.P(Jan-Mar) = P(Apr-Jun) = P(Jul-Sep) = P(Oct-Dec)At least one of the probabilities is different.P(Jan-Mar) ≠P(Apr-Jun) ≠P(Jul-Sep) ≠P(Oct-Dec)

P(Oct-Dec) is less than the other probabilities.

P(Jul-Sep) is greater than the other probabilities.At least one of the probabilities is different.P(Jan-Mar) ≠P(Apr-Jun) ≠P(Jul-Sep) ≠P(Oct-Dec)P(Oct-Dec) is less than the other probabilities.

P(Jan-Mar) = P(Apr-Jun) = P(Jul-Sep) = P(Oct-Dec)


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(5) P(Jan-Mar) ≠P(Apr-Jun) ≠P(Jul-Sep) ≠P(Oct-Dec).P(Jul-Sep) is greater than the other probabilities for each quarter at this college.P(Jan-Mar) = P(Apr-Jun) = P(Jul-Sep) = P(Oct-Dec).at least one of the probabilities of birthdays in each quarter at this college is different.

P(Oct-Dec) is less than the other probabilities for each quarter at this college.


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1. B. : The death rate has remained the same at % after starting hand-washing.H0 14.2

: The death rate has decreased to a value less than %.HA 14.2

C. : p , : p (p is the proportion of deaths at the clinic.)H0 = 0.142 HA < 0.142

2. (1) p

(2) =

0.34

(3) p

(4) >

0.34

(5) right-tailed

3. (1) =

0.50

(2) <

0.50

− 7.72

0.000

(3) Reject

(4) less than

B. The proportion %.is significantly less than 50

4. C. : Obesity and relationship status are independent.H0 : Obesity and relationship status are related.HA

29.52

0.000

D. and conclude that obesity and relationship status related.Reject H0 are

B. No. Although a relationship is evident, this is an observational study and cannot be used to prove causality.

5. C. The null hypothesis always states that there is no relationship between the variables.

6. C. : Gender and happiness are independentH0 : Gender and happiness are dependentHa

A. and conclude that gender and happiness independent.Fail to reject H0 could be


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7. 891

232

1123

1272

215

1487

930.67

192.33

1232.33

254.67

C. There are more women who believe in the afterlife and men who do not than is expected.

17.3

8. 61.7

38.3

5.4

94.6

C.The data suggest that marijuana use is much more common for those who have smoked cigarettes than for those who have not.

C.There are two categorical variables, randomization was used to obtain the data, and the expected count was at least five in all cells.

B. : Cigarette use and marijuana use are independent.H0 : Cigarette use and marijuana use are dependent.Ha

652

0.0000

D. and conclude that the two variables are dependent.Reject H0

9. A.This is false. The steps of the test and the results are identical no matter which of the two categorical variables defines the rows or columns.

20.22

0.000


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10. 31

659

690

20

660

680

51

1319

A.There are two categorical variables, randomization was used to obtain the data, and the expected count was at least five in all cells.

C. : Taking aspirin and having a heart attack are independent.H0: Taking aspirin and having a heart attack are dependent.Ha

2.300

D. 0.1293

C. and conclude that the two variables .Fail to reject H0 could be independent

11. C. versus H : p = 0.750 H : p ≠ 0.75a

3.3

1

0.07

A.The probability of obtaining a test statistic as extreme as that observed, assuming the null hypothesis is true, is . There is evidence against the null hypothesis, but not very strong.

0.07

12. (1) P(Jan-Mar) = P(Apr-Jun) = P(Jul-Sep) = P(Oct-Dec)

(2) At least one of the probabilities is different.

21.500

21.500

21.500

21.500

8.791

3

0.032

(3) Reject

(4) is

(5) at least one of the probabilities of birthdays in each quarter at this college is different.


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