Experience 13 – Goodness Of Fit (GOF) Tests

• You are conducting a multinomial Goodness of Fit hypothesis test for the claim that the 4 categories occur with the following frequencies:

You are conducting a multinomial hypothesis test (?? = 0.05) for the claim that all 5 categories are equally likely to be selected. Complete the table.

Category	Observed Frequency	Expected Frequency
A	8
B	15
C	5
D	19
E	17

Report all answers accurate to three decimal places. But retain unrounded numbers for future calculations.

What is the chi-square test-statistic for this data? (Report answer accurate to three decimal places, and remember to use the unrounded Pearson residuals in your calculations.)
?2=?2=

What are the degrees of freedom for this test?
d.f.=

What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value =

The p-value is…

• less than (or equal to) ??
• greater than ??

This test statistic leads to a decision to…

• reject the null
• accept the null
• fail to reject the null
• accept the alternative

As such, the final conclusion is that…

• There is sufficient evidence to warrant rejection of the claim that all 5 categories are equally likely to be selected.
• There is not sufficient evidence to warrant rejection of the claim that all 5 categories are equally likely to be selected.
• The sample data support the claim that all 5 categories are equally likely to be selected.
• There is not sufficient sample evidence to support the claim that all 5 categories are equally likely to be selected.

HoHo : pA=0.3pA=0.3; pB=0.1pB=0.1; pC=0.2pC=0.2; pD=0.4pD=0.4

Complete the table. Report all answers accurate to three decimal places.

Category	Observed Frequency	Expected Frequency
A	41
B	29
C	13
D	51

What is the chi-square test-statistic for this data? (2 decimal places)
?2=?2=

What is the P-Value? (3 decimal places)
P-Value =

For significance level alpha 0.005,

What would be the conclusion of this hypothesis test?

• Reject the Null Hypothesis
• Fail to reject the Null Hypothesis

Report all answers accurate to three decimal places.

Critical Thinking E13

Questions:

• What are the requirements to perform a goodness-of-fit test? [2 bullets]
• Why is the chi-square distribution always positive? [1 sentence]
• How would the goodness-of-fit procedure be modified to check that data follows a continuous distribution? [2 sentences]

Applications E13

You might think that if you looked at the first digit in randomly selected numbers that the distribution would be uniform. Actually, it is not! Simon Newcomb and later Frank Benford both discovered that the digits occur according to the following distribution: (digit, probability)

(1,0.301), (2,0.176),(3,0.125),(4,0.097),(5,0.079),(6,0.067),(7,0.058),(8,0.051),(9,0.046)

The IRS currently uses Benford’s Law to detect fraudulent tax data. Suppose you work for the IRS and are investigating an individual suspected of embezzling. The first digit of 155 checks to a supposed company are as follows:

Digit	1	2	3	4	5	6	7	8	9
Observed Frequency	43	31	17	18	21	9	14	1	1

a. State the appropriate null and alternative hypotheses for this test.

b. Explain why ?=0.01 is an appropriate choice for the level of significance in this situation.

c. What is the P-Value? Report answer to 4 decimal places
P-Value =

d. What is your decision?

• Fail to reject the Null Hypothesis
• Reject the Null Hypothesis

e. Write a statement to the law enforcement officials that will use it to decide whether to pursue the case further or not. Structure your essay as follows:

Given a brief explanation of what a Goodness of Fit test is.
Explain why a Goodness of Fit test should be applied in this situation.
State the hypotheses for this situation.
Interpret the answer to part c.
Use the answer to part c to justify the decision in part d.
Use the decision in part d to make a conclusion about whether the individual is likely to have embezzled.
Use this to then tell the law enforcement officials whether they should pursue the case or not.

Experience 14 – Linear Correlation

Annual high temperatures in a certain location have been tracked for several years. Let XX represent the year and YY the high temperature. Based on the data shown below, calculate the regression line (each value to two decimal places).

y = x +

x	y
5	17.2
6	15.68
7	16.76
8	19.64
9	19.72
10	23.8
11	21.98
12	25.96
13	23.84
14	25.92
15	28.9
16	27.08
17	27.66
18	28.44
19	33.42
20	32.8

A researcher wishes to examine the relationship between years of schooling completed and the number of pregnancies in young women. Her research discovers a linear relationship, and the least squares line is: ˆy=3?5xy^=3-5xwhere x is the number of years of schooling completed and y is the number of pregnancies. The slope of the regression line can be interpreted in the following way:

• When amount of schooling increases by one year, the number of pregnancies increases by 5.
• When amount of schooling increases by one year, the number of pregnancies decreases by 5.
• When amount of schooling increases by one year, the number of pregnancies increases by 3.

When amount of schooling increases by one year, the number of pregnancies decreases by 3.

Annual high temperatures in a certain location have been tracked for several years. Let XX represent the year and YY the high temperature. Based on the data shown below, calculate the correlation coefficient (to three decimal places) between XX and YY. Use your calculator!

x	y
2	19
3	15.97
4	13.84
5	14.21
6	12.28
7	9.65
8	9.02
9	8.49
10	4.56
11	4.73
12	1.8

r=

Annual high temperatures in a certain location have been tracked for several years. Let XX represent the year and YY the high temperature. At the 0.05 significance level, does the data below show significant (linear) correlation between XX and YY?

x	y
2	15.68
3	17.57
4	22.76
5	28.95
6	30.34
7	23.53
8	35.02
9	30.91
10	35.1
11	39.89
12	40.08
13	37.77
14	36.86

• Yes, significant correlation
• No

Applications E14

Bone mineral density and cola consumption has been recorded for a sample of patients. Let xx represent the number of colas consumed per week and yy the bone mineral density in grams per cubic centimeter. Based on the data shown below answer the questions rounding your final answers to four decimal places.

(a) Create a scatter plot with linear regression line for the data. (2 points)

y = x +

(b) Interpret the slope of the regression equation in a complete sentence. (2 points)

(c) Use the linear correlation coefficient to determine if there is correlation. (5 points)

r =

Is there correlation at the 0.05 level of significance? (2 points)

• yes
• no

(d) According to the linear regression equation, the bone density of someone who drinks 24 colas per week is . (2 points)

Submit a file with your scatter plot below. (5 points)

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