# Part i. chi-square goodness of fit test (equal frequencies) in order

Part I. Chi-Square Goodness of Fit Test (equal frequencies)

In order to determine staffing levels, a hospital wants to determine if births occur with the same frequency on each day of the week. The table below lists the days of the week selected by a random sample of 100 births. Consider the claim that the days of the week have the same frequency of a birth occurring.

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

32

16

7

20

11

8

6

Instructions for performing this test in Stat Disk can be found in the Stat Disk User’s Manual under Goodness of Fit, Equal Frequencies.

Instructions

1. Use   the Chi-Square Goodness-of-Fit test to see if there is a difference between   the frequency of births and days of the week.  Use a significance level of .01

Paste results here.

2. What   are we trying to show here?

3. What   is the p-value and what does it represent in the context of this problem?

4.  State in your own words what the results of   this Goodness-of-fit test tells us.

5. Repeat   the above procedure using only the weekdays Paste results here. Did you get different results? What do they mean?

Part II. Chi-Square Goodness of Fit Test (unequal frequencies)

In the 2000 U.S. Census, the ages of individuals in a small town were found to be the following:

Less than 18 – 20% 18-35 – 30%  and  greater than 35 – 50%

In 2010 the ages of 500 individuals from the same small town were sampled with the following results:

Less than 18 – 121 people 18-35 – 288 people and greater than 35 – 91 people

Using an alpha of 0.05, would you conclude that the population distribution of ages has changed in the last 10 years? Instructions for performing this test in Stat Disk can be found in the Stat Disk User’s Manual under Goodness of Fit, Unequal Frequencies.

Instructions

6. Complete   the table as necessary.

[Hint: You will   need to compute the observed frequencies based on the percentages for the 500   samples. Round to the nearest   integer.]

Less than 18

18-35

35-91

OBSERVED

121

288

91

EXPECTED

7. Use   the Chi-Square Goodness-of-Fit test for Unequal frequencies to see if there   is a difference between the observed frequencies and the expected frequencies   Use a significance level of .05

Paste results here.

8. State   the null and alternative hypothesis.

9. What   conclusion would you reach, given the result of your Goodness-of-Fit   test? [State in your own words   following the guidelines for a conclusion statement learned last week.]

Part III. Chi-Square Test of Independence

The following table is the result of a survey from a random sample of different crime victims.

Use the data to test the claim that the type of crime is independent of whether the criminal was a stranger. Use a significance level of 0.05.

Homicide

Robbery

Assault

Criminal was a   stranger

12

379

727

Criminal was an   acquaintance or relative

39

106

642

Hint: Instructions for performing this test in Stat Disk can be found in the Stat Disk User’s Manual under the heading Chi Square Test of Independence (Contingency Tables).

Instructions

10. Just   looking at the numbers in the table, what is your best guess about the   relationship between type of crime and criminal being a stranger or not? Are   they independent or is there a relationship?

11. Compute a   Chi-Square Test of Independence on this data using a 0.05 level of significance. Paste your results here.

12. What are   the null and alternative hypothesis for this test?

13. What is   the p-value for this result? What does   this represent?

14. State   your conclusion related to the context of this problem.

Part IV. Apply this to your own situation

Using one of the above statistical tests, compose and SOLVE an actual problem from the context of your own personal or professional life. You will need to make up some data and describe which test you will use to analyze the situation. Here’s an example:

Example: Do not use this   problem!!

State the problem that you are analyzing.

Last year, I asked the kids in my neighborhood what   kind of cookies they preferred. 50%   said chocolate-chip, 20% said oatmeal-raisin, and 30% said sugar cookie. I want to see if this has changed.

Make up some data for the new situation.

I asked 50 neighborhood kids what kind of cookie they   preferred now and here’s what they said:

· 35 said chocolate-chip

· 5 said oatmeal-raisin

Determine which type of Chi-Square test you will   perform.

Since these are unequal frequencies, I will perform a   Chi-Square Goodness-of-Fit Test (Unequal Frequencies).

Specify your null and alternative hypotheses.

H0: There is no   difference this year in the preferences of cookies within the neighborhood   kids.

H1: Things have   changed.

Setup the test

Chocolate-Chip

Oatmeal-Raisin

OBSERVED

35

5

10

EXPECTED

25

10

15

Perform the test

We have   evidence to believe ….

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Part I. Chi-Square Goodness of Fit Test (equal frequencies)

In order to determine staffing levels, a hospital wants to determine if births occur with the same frequency on each day of the week. The table below lists the days of the week selected by a random sample of 100 births. Consider the claim that the days of the week have the same frequency of a birth occurring.

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

32

16

7

20

11

8

6

Instructions for performing this test in Stat Disk can be found in the Stat Disk User’s Manual under Goodness of Fit, Equal Frequencies.

Instructions

1. Use   the Chi-Square Goodness-of-Fit test to see if there is a difference between   the frequency of births and days of the week.  Use a significance level of .01

Paste results here.

2. What   are we trying to show here?

3. What   is the p-value and what does it represent in the context of this problem?

4.  State in your own words what the results of   this Goodness-of-fit test tells us.

5. Repeat   the above procedure using only the weekdays Paste results here. Did you get different results? What do they mean?

Part II. Chi-Square Goodness of Fit Test (unequal frequencies)

In the 2000 U.S. Census, the ages of individuals in a small town were found to be the following:

Less than 18 – 20% 18-35 – 30%  and  greater than 35 – 50%

In 2010 the ages of 500 individuals from the same small town were sampled with the following results:

Less than 18 – 121 people 18-35 – 288 people and greater than 35 – 91 people

Using an alpha of 0.05, would you conclude that the population distribution of ages has changed in the last 10 years? Instructions for performing this test in Stat Disk can be found in the Stat Disk User’s Manual under Goodness of Fit, Unequal Frequencies.

Instructions

6. Complete   the table as necessary.

[Hint: You will   need to compute the observed frequencies based on the percentages for the 500   samples. Round to the nearest   integer.]

Less than 18

18-35

35-91

OBSERVED

121

288

91

EXPECTED

7. Use   the Chi-Square Goodness-of-Fit test for Unequal frequencies to see if there   is a difference between the observed frequencies and the expected frequencies   Use a significance level of .05

Paste results here.

8. State   the null and alternative hypothesis.

9. What   conclusion would you reach, given the result of your Goodness-of-Fit   test? [State in your own words   following the guidelines for a conclusion statement learned last week.]

Part III. Chi-Square Test of Independence

The following table is the result of a survey from a random sample of different crime victims.

Use the data to test the claim that the type of crime is independent of whether the criminal was a stranger. Use a significance level of 0.05.

Homicide

Robbery

Assault

Criminal was a   stranger

12

379

727

Criminal was an   acquaintance or relative

39

106

642

Hint: Instructions for performing this test in Stat Disk can be found in the Stat Disk User’s Manual under the heading Chi Square Test of Independence (Contingency Tables).

Instructions

10. Just   looking at the numbers in the table, what is your best guess about the   relationship between type of crime and criminal being a stranger or not? Are   they independent or is there a relationship?

11. Compute a   Chi-Square Test of Independence on this data using a 0.05 level of significance. Paste your results here.

12. What are   the null and alternative hypothesis for this test?

13. What is   the p-value for this result? What does   this represent?

14. State   your conclusion related to the context of this problem.

Part IV. Apply this to your own situation

Using one of the above statistical tests, compose and SOLVE an actual problem from the context of your own personal or professional life. You will need to make up some data and describe which test you will use to analyze the situation. Here’s an example:

Example: Do not use this   problem!!

State the problem that you are analyzing.

Last year, I asked the kids in my neighborhood what   kind of cookies they preferred. 50%   said chocolate-chip, 20% said oatmeal-raisin, and 30% said sugar cookie. I want to see if this has changed.

Make up some data for the new situation.

I asked 50 neighborhood kids what kind of cookie they   preferred now and here’s what they said:

· 35 said chocolate-chip

· 5 said oatmeal-raisin

Determine which type of Chi-Square test you will   perform.

Since these are unequal frequencies, I will perform a   Chi-Square Goodness-of-Fit Test (Unequal Frequencies).

Specify your null and alternative hypotheses.

H0: There is no   difference this year in the preferences of cookies within the neighborhood   kids.

H1: Things have   changed.

Setup the test

Chocolate-Chip

Oatmeal-Raisin

OBSERVED

35

5

10

EXPECTED

25

10

15

Perform the test

We have   evidence to believe ….

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