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# Hypothesis testing problems with chi-squared, ANOVA and t tests

1. You would like to help some of your math-deficient friends understand probability before they loose too much on lottery tickets. After shuffling two decks of cards, you lay down 40 of them. You say that there should be 10 in each of the four suits: hearts, diamonds, spades, and clubs. What you actually get is 9 hearts, 8 diamonds, 10 spades, and 13 clubs. Close enough you say. What are the factors to be researched? What level of data is described? What statistic would you use to make your decision?

2. The mental health and rehabilitation counselors at one state agency were fighting over who was getting paid the most. A mediator was brought in to study the situation and tell the decision makers if a significant difference was found between the groups. What are the factors to be researched? What level of data is described? What statistic would you use to make your decision?

3. You have been fortunate enough to receive a grant from a foundation studying pain in arthritis victims under the age of 30. You have not found a lot of people, but you think that you may have identified something that works pretty good. They are measured on an established scale pain scale from 20 (not much pain) to 100 (a lot of pain). You had already started treating them in two different ways before you realized, "Boy-howdy, I should have taken pain measurements at the beginning." All you can do now is a measurement on both groups at the end of three months, but you can do that. Assuming that they all started with about the same scores, here's the pain scores at the end for your six patients, three in each group.

Heat/stretching acupuncture/deep massage/diet

45 34
50 29
38 41

a. Write null hypothesis for this research.
b. What is the level of measurement used in this research?
c. Are the two groups being measured the same or independent?
d. What inferential statistic would you recommend to determine if the difference between the two groups is significant?
f. Are the two groups significantly different from each other? (Show your work.)
g. Write a summary statement.

4. In some places in the US, people are aware of the need for more care for older citizens, like Dr. Mobley. In the election in one area, they put forward a referendum that would increase the county sales tax by one cent to pay for increased services for this important part of the community. Registered voters were asked to vote Yes, raise the tax to provide more services to people 60 and older, or No, do not raise the tax to provide more services to people 60+, on the referendum. Fortunately, for the younger people of the county who would pay most of the tax, it was defeated. The discussion that followed included a vigorous rehashing of the pro's and con's of the referendum. You accidentally were in attendance one night and had an idea: what was the idea and how do you do it?

Hint: if there was a significant difference between those voting Yes compared with the No's, then . . . If the Yes vote was 1001 and No was 1258, do you have an argument?

a. Write null hypothesis for this research.
b. What is the level of measurement used in this research?
c. What assumptions are you making?
d. Are the two groups being measured the same or independent?
e. What statistic would you recommend to determine if the difference between the two groups is significant?
f. Are the two groups significantly different from each other? (Show your work.)
g. Extra credit: what is the lowest number of No votes it would take in this situation for there to be a significant difference? (Show work.)
h. Write a summary statement.

5. Some feminists have asserted that the Wechsler IQ tends to treat females differently from males. Another group, who advocates for people with psychiatric disorders, might say the same thing about people with psychiatric disorders. To examine these alleged differences they studied the following clients using only the most indicative Wechsler scale score, which they recorded.

Diagnosis 1 Diagnosis 2 Diagnosis 3
Shunica 10 Fran 9 Edith 12
Ed 6 Tom 12 Paul 10
Reggie 15 Mike 10 Ed2 13
Hope 4 Ashlie 7 Carla 13
Laura 13 Megan 10 Kita 15
Reggie2 12 David 12 Paul2 15
Marsha 9 Merita 7 Trish 10
Troy 12 Daniel 13 Michael 12
Julius 13 Herbert 7 James 10
Maria 5 Dawn 13 Marla 13

a. Write null hypothesis/hypotheses for this research.
b. What is the level of measurement used in this research?
c. Are the groups being measured the same or independent?
d. What inferential statistic would you recommend to determine if the difference between the groups is significant?
e. Are the groups significantly different from each other? (Use the computer and print the results.)
f. Write a summary statement.

See attached file for full problem description.

#### Solution Preview

1. You would like to help some of your math-deficient friends understand probability before they loose too much on lottery tickets. After shuffling two decks of cards, you lay down 40 of them. You say that there should be 10 in each of the four suits: hearts, diamonds, spades, and clubs. What you actually get is 9 hearts, 8 diamonds, 10 spades, and 13 clubs. Close enough you say.

What are the factors to be researched?
What level of data is described?
What statistic would you use to make your decision?

Here we can consider the expected and observed number of cards from each suit. The expected number of cards are 10 each from hearts, diamonds, spades and clubs

Suit Expected Observed
Heart 10 9
diamond 10 8
Club 10 13

Here the level of measurement is nominal.

Here we can use Chi Square test to test whether the Expected and Observed frequencies are significantly differ. Chi square test statistic is given by

Rejection criteria: Reject the null hypothesis if the calculated value of Chi square is grater than the critical value with 3 d.f ( at 5% significance level)

Critical value = 7.81
Details

Suit Expected -E Observed -O (O-E)2 / E
Heart 10 9 1/10
Diamond 10 8 4/10
Club 10 13 9/10
Total 40 40 14/10

Calculated value = 1.4
Critical value = 7.81

Conclusion: Accept the null hypothesis that there is no significant difference in the observed and expected number of cards for the above experiment

2. The mental health and rehabilitation counselors at one state agency were fighting over who was getting paid the most. A mediator was brought in to study the situation and tell the decision makers if a significant difference was found between the groups. What are the factors to be researched? What level of data is described? What statistic would you use to make your decision?

To examine whether there is any significant difference between mental health and rehabilitation counselors, the following factors studied.

Factors Level of measurement
1. Salary Ratio
2. Working hours Ratio
3. Years of Experience Ratio
4. Educational Qualifications Ordinal
5. Job Category Ordinal

Ratio type measurements can be compared using Student's t test. While Ordinal type measurements can be compared using chi square test.

3. You have been fortunate enough to receive a grant from a foundation studying pain in arthritis victims under the age of 30. You have not found a lot of people, but you think that you may have identified something that works pretty good. They are ...

#### Solution Summary

The solution contain hypothesis testing problems related to ANOVA, Students t and Chi-square test. A detailed description of Null Hypothesis, Alternative Hypothesis, Students t test, One sample t test, decision rule, Confidence level, Independent sample t test and Test Statistic are made with interpretations of the results.

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