PSY-380 Introduction to Probability and Statistics

Project 3

Answer each question completely. Copy and Paste the SPSS output into the word document for the calculations portion of the problems, highlighting correct answers. (Please remember to answer the questions you must interpret the SPSS output).

A researcher is interested to learn if there is a linear relationship between the hours in a week spent exercising and a person’s life satisfaction. The researchers collected the following data from a random sample, which included the number of hours spent exercising in a week and a ranking of life satisfaction from 1 to 10 ( 1 being the lowest and 10 the highest).

Participant

Hours of Exercise

Life Satisfaction

1

3

1

2

14

2

3

14

4

4

14

4

5

3

10

6

5

5

7

10

3

8

11

4

9

8

8

10

7

4

11

6

9

12

11

5

13

6

4

14

11

10

15

8

4

16

15

7

17

8

4

18

8

5

19

10

4

20

5

4

Find the mean hours of exercise per week by the participants. PSY-380 Introduction to Probability and Statistics Project 3

Find the variance and standard deviation of the hours of exercise per week by the participants.

Run a bivariate correlation to determine if there is a linear relationship between the hours of exercise per week and the life satisfaction. Report the results of the test statistic using correct APA formatting.

Run a linear regression on the data. Report the results, using correct APA formatting. Identify the amount of variation in the life satisfaction ranking that is due to the relationship between the hours of exercise per week and the life satisfaction (Hint: the R2 value)

Report a model of the linear relationship between the two variables using the regression line formula.

Insomnia has become an epidemic in the United States. Much research has been done in the development of new pharmaceuticals to aide those who suffer from insomnia. Alternatives to the pharmaceuticals are being sought by sufferers. A new relaxation technique has been tested to see if it is effective in treating the disorder. Sixty insomnia sufferers between the ages of 18 to 40 with no underlying health conditions volunteered to participate in a clinical trial. They were randomly assigned to either receive the relaxation treatment or a proven pharmaceutical treatment. Thirty were assigned to each group. The amount of time it took each of them to fall asleep was measured and recorded. The data is shown below. Run an independent samples t-test to determine if the relaxation treatment is more effective than the pharmaceutical treatment at a level of significance of 0.05. Report the test statistic using correct APA formatting and interpret the results.

Relaxation

Pharmaceutical

98

20

117

35

51

130

28

83

65

157

107

138

88

49

90

142

105

157

73

39

44

46

53

194

20

94

50

95

92

161

112

154

71

75

96

57

86

34

92

118

75

41

41

145

102

148

24

117

96

177

108

119

102

186

35

22

46

61

74

75

A researcher is interested to learn if the level of interaction a women in her 20s has with her mother influences her life satisfaction ratings. Below is a list of women who fit into each of four levels of interaction. Conduct a One-way ANOVA on the data to determine if there are differences between groups; does the level of interaction influence women’s ratings of life satisfaction? Report the results of the One-way ANOVA. If significance is found, run the appropriate post-hoc test and report between what levels the significant differences were found. Report the test statistic using correct APA formatting and interpret the results. PSY-380 Introduction to Probability and Statistics Project 3

No Interaction

Low Interaction

Moderate Interaction

High Interaction

2

3

3

9

4

3

10

10

4

5

2

8

4

1

1

5

7

2

2

8

8

2

3

4

1

7

10

9

1

8

8

4

8

6

4

1

4

5

3

8

Are handedness and gender related? A researcher collected the following data in hopes of discovering if handedness and gender are independent (Ambidextrous individuals were excluded from the study). Use the Chi-Square test for independence to explore this at a level of significance of 0.05. Report the test statistic using correct APA formatting and interpret the results.

Left-Handed

Right-Handed

Men

13

22

Women

27

18

A researcher is interested in studying the effect that the amount of fat in the diet and amount of exercise has on the mental acuity of middle-aged women. The researcher used three different treatment levels for the diet and two levels for the exercise. The results of the acuity test for the subjects in the different treatment levels are shown below.

Diet

Exercise

<30% fat

30% – 60% fat

>60% fat

<60 minutes

4

3

2

4

1

2

2

2

2

4

2

2

3

3

1

60 minutes

6

8

5

or more

5

8

7

4

7

5

4

8

5

5

6

6

Perform a Two-way analysis of variance (ANOVA) and report the results using correct APA style; report whether significance was found for Factor A, Factor B, and/or an interaction between Factors A and B was found.

If the test statistic is significant, run a post hoc test to determine between what groups significance was found.

Report an effect size for all significant results. PSY-380 Introduction to Probability and Statistics Project 3

 

 

MORE INFO 

Introduction to Probability and Statistics

Introduction

Probability is the branch of mathematics that deals with random events. The word “probability” comes from two Latin words, probare and pro babilis, meaning “to test or to prove.” Probability can be thought of as how likely something is to happen or as a measure of uncertainty about an event or sequence. The term “statistics” refers both to the science of collecting numerical data about things and also to its application in reasoning about those data.

Random Variables

A random variable is a variable that can take on any value from a set of possible values. For example, the weight of an object might be described as being between 60 and 80 pounds. We could also say that this is a random variable because its possible outcomes are not fixed; there are many different values that it could take on.

The sample space consists of all possible outcomes for our random variable—the same thing as saying “all possible values” or “all possible results” (like we did earlier when talking about probabilities). It’s just like having one big bowl full of different types of cookies: if you wanted to know how many cookies there were in total, you’d have to sample them all before finding out their total number!

So what exactly does sampling mean? In statistics terms, it means taking some data points from your population and seeing what kind of relationship they have with each other!

Normal Distribution

The normal distribution is a continuous probability distribution, which means that it can take on any value between 0 and 1. This is because the mean, or center of gravity, of your dataset will always be 0 and its standard deviation will always be 1.

The normal distribution has an average value of 0 and a standard deviation (or “standard error”) equal to 1. The curve itself looks like an upside-down bell shape with its peak at around −0.5 standard deviations from the mean (representing half as many units away from center).

The most common use for this model is predicting what percentage of data points should fall above or below some threshold (like when you need to know how many people have never been married before). You can use this relationship between two variables—for example, comparing how many people are under 30 versus over 60—to estimate how much older someone must be before being considered eligible for Medicare benefits based on their age alone.

Mean and Standard Deviation

Mean and standard deviation are two of the most important statistics in probability.

Mean is the average value of a set of data points, while standard deviation is a measure of how spread out those data points are. The mean tells us how many times an observation (a single number) falls above or below another observation; this can be used to describe how close one number is to another on average. For example, if you have three numbers that represent temperatures—20°C, 30°C and 40°C—and you want to find out what their average temperature is (their mean), you would sum up all their values together: 20 + 30 + 40 = 80°C. However if we instead looked at these values individually as deviations from each other instead of adding them up as averages we could determine whether any particular number was larger than or smaller than others by looking at its absolute value rather than just taking its distance from its neighbours’ means: 20 – 12 = 6; 30 – 9 = 15; 40 – 7 = 22

The standard deviation tells us how far off each individual value may fall compared with its neighbour(s). It cannot tell us whether something will be bigger than anything else though since it only measures absolute deviations from other numbers within one dataset – which doesn’t help much when comparing different datasets!

Covariance and Correlation

Covariance and correlation are two measures of the linear relationship between variables. Covariance is a measure of how two variables change together. Correlation is a measure of how two variables change together. In both cases, you can think of it as if you had one variable that was measuring something else (like temperature) and another variable that measured temperature as well (like humidity). If the humidity increases and decreases at the same time as your temperature changes, then we would say there’s positive covariance between these two variables—that is, their values tend to move in tandem over time because they’re related somehow!

Correlation doesn’t tell us anything about whether or not there’s actually any kind of causation going on here: maybe your body always feels cold when its internal environment gets colder outside because it has adapted over evolutionary history so much that now being cold makes our bodies feel more comfortable than feeling warm does—but still no matter what happens outside our bodies will never get hot enough for us to sweat through clothes during summertime heat waves… But if we see lots more days where both temperature and humidity go up/down at exactly similar rates… I’m going out tonight after work 🙂

Central Limit Theorem

The central limit theorem states that the mean of a sample from a population that has a normal distribution will be approximately normally distributed. This means that if we take many samples from the same population and plot their means, we will find them to be clustered around the mean.

The use of this theorem can be helpful for two reasons:

• It provides us with an estimate of our sample size needed to achieve adequate power (refer back to p = 0.80) when we want to detect differences between groups using one-sample t tests or two-sample Wilcoxon rank sum test statistics.* It can also help us determine whether our data are consistent with an approximation made by assuming normality in order for our results not only look “right,” but also fit well with what would happen if they were truly random samples drawn from populations with known distributions

Baye’s Theorem and Applications

Baye’s Theorem is used to find the probability of an event. It’s a theorem in probability theory and can be used in statistics.

Baye’s Theorem states that if you have two random variables X and Y, then:

• P(X=x)=P(Y=y) = P(XY)=P(X|Y=y).

• In other words, we can use Bayes’ theorem to find out how likely it is for one thing to happen based on what we already know about another thing (like the weather or our likelihood of winning a lottery ticket). This can be useful if you want to predict something by analyzing data from previous events; however, there are some caveats when using this method.

Takeaway:

Probability is a branch of mathematics that studies chance and randomness in real-world situations.

In this lesson, you will learn about probability and statistics: how they work together to help us understand the world around us. We’ll start by learning about what probabilities are and why we need them. Then we’ll explore some basic concepts like probability distributions, statistical significance and hypothesis testing. Finally, we’ll talk about how to apply these concepts in your everyday life as well as in your career field!

Conclusion

Probability is an important part of statistics, and it’s one of the foundations of modern probability theory. It can be tricky to get your head around at first, but once you understand the basics we’ve discussed here, it will be easier to see why random variables are so useful in statistics!