Please see the attached file.

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?

Answer

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
Spade 10 10
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
Spade 10 10 0/10
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?

Answer

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 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.

Traditional Traditiona
Heat/stretching acupuncture/deep massage/diet

45 34
50 29
38 41

a. Write null hypothesis for this research.
H0: There is no significant difference in the mean pain sale of the two groups

b. What is the level of measurement used in this research?

Here the measurement used is pain scale score which can be treated as an Interval scale measurement. In interval measurement the distance between attributes does have meaning.

c. Are the two groups being measured the same or independent?

The groups are independent.

d. What inferential statistic would you recommend to determine if the difference between the two groups is significant?

Here we can use independent sample t test to check whether there is any difference between the two groups

e. Use another statistic and compare your answers.

Traditional Nontraditional
Heat/stretching acupuncture/deep massage/diet

38 29
45 34 Median
50 41

We cam also use median to compare the two groups. Here the median for traditional heat / stretching is higher than nontraditional acupuncture / deep massage / diet

f. Are the two groups significantly different from each other? (Show your work.)

t-Test: Two-Sample Assuming Equal Variances
Traditional Non Traditional
Mean 44.33333333 34.66667
Variance 36.33333333 36.33333
Observations 3 3
Pooled Variance 36.33333333
Hypothesized Mean Difference 0
df 4
t Stat 1.964127838
P(T<=t) one-tail 0.060490392
t Critical one-tail 2.131846782
P(T<=t) two-tail 0.120980784
t Critical two-tail 2.776445105

Since the calculated value of t (1.964127) is less than the critical value 2.77644, we accept the null hypothesis that there is no significant difference in the mean pain scale of two groups

g. Write a summary statement.
We accept the null hypothesis that there is no significant difference in the mean pain scale of two groups

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.
H0: There no significant difference between those voting Yes compared with the No's,

b. What is the level of measurement used in this research?
The level of measurement used here is ordinal

c. What assumptions are you making?
The assumption is that the two groups are disjoint.
d. Are the two groups being measured the same or independent?

The two groups are independent.

e. What statistic would you recommend to determine if the difference between the two groups is significant?

Here Z test can be used to check whether the two proportion are equal or not
The Z Statistic is given by

f. Are the two groups significantly different from each other? (Show your work.)

Given that n1= n2= 2259 ,
p1 = 1001/2259 = 0.443116
p2 = 1258 / 2259=0.556884

Calculated value of Z = -7.647
Critical value of Z at 5% Significance level = ± 1.96
P value = 0.000

Since the calculated value of Z is greater than the critical value, we reject the null hypothesis

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.)

The 95 % Confidence interval of P1-P2 is given by

where

Thus the lowest number of No votes it would take in this situation for there to be a significant difference is given by
(n1+n2)* ( ) = 2259*0.515376=1164.235

h. Write a summary statement.

The number of votes in favor of referendum is significantly lower than the votes against it.

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.

Null Hypothesis

H0: There is no significant difference in the mean Wechsler scale score under the three diagnosis

H1: There is significant difference in the mean Wechsler scale score under the three diagnosis.

b. What is the level of measurement used in this research?

Here the measurement used is Wechsler scale score which can be treated as an Interval scale measurement. In interval measurement the distance between attributes does have meaning.

c. Are the groups being measured the same or independent ?

Since different persons are involved in the three types of Diagnosis , they can be treated as independent

d. What inferential statistic would you recommend to determine if the difference between the groups is significant?

Here F Statistic from Analysis of Variance (ANOVA) technique can be used to examine whether the difference between the groups is significant.

e. Are the groups significantly different from each other? (Use the computer and print the results.)

SUMMARY
Groups Count Sum Average Variance
diagnosis.1 10 99 9.9 14.32222222
diagnosis.2 10 100 10 6
diagnosis.3 10 123 12.3 3.566666667

ANOVA
Source of Variation SS df MS F P-value F crit
Between Groups 36.86667 2 18.43333 2.314883721 0.118063 3.354131
Within Groups 215 27 7.962963
Total 251.8667 29

Here the calculated value of test statistic is 2.3148 which is less than the critical value 3.354131 at 5% significance level. Thus we accept the null hypothesis that there is no significant difference.

f. Write a summary statement.

The ANOVA test suggests that there is no significant difference the mean Wechsler scale score under the three diagnosis.