## Stochastics

Problem 1 [10 points]
Random variables, fTj : j 1g are independent with a common exponential density function,
g(t) = exp( t) for t > 0;
with = 5 per hour. Introduce the sums,
k
X
Wk =
Tj and W0 = 0:
j=1
Consider a process, N = fN(t) : t 0g de ned as follows:
[N(t) = n] () [Wn t < Wn+1]
1. Derive E [W1 jW3 1 < W4]
2. Evaluate expectation E [W1 jW4 = 2]
Solution
2
Problem 2 [10 points]
Random variables, fTj : j 1g are independent with a common exponential density function, g(t) =
exp( t) for t > 0; with = 5 per hour. Introduce the sums,
k
X
Wk =
Tj and W0 = 0:
j=1
Consider a process, N = fN(t) : t 0g de ned as follows: [N(t) = n] () [Wn t < Wn+1]
1. Derive expectation of W5; given that W2 = 1 (in hours).
2. Evaluate expectation of the ratio, (W5=W2)
Solution
3
Problem 4 [10 points]
Consider a small service with arrivals described as a Poisson process, N = fN(t) : t 0g such that the
rst arrival time, W1 = S1; has E [S1jN(0) = 0] = 6 minutes, or (0.1) of an hour.
1. Find conditional expected value for a number of customers arrived by the end of rst hour, given
that by t = 3 hours there were ten customers.
2. Evaluate expected number of customer by t = 3 hours, given that by the end of rst hour there were
four customers.
Solution
5
Problem 5 [10 points]
Consider a queuing system, M=M=1 with one server and parameters such that customer arrivals are
described by a Poisson process with = 3 per hour, and service times are independent exponentially
distributed with
1
= 5 minutes.
1. Derive the average queue length, E [X(t)]; assuming that the process X = fX(t) : t 0g follows the
stationary distribution.
2. Evaluate expected busy time.
Solution
6
Problem 10 [10 points]
Consider a Poisson process, N = fN(t) : t 0g with rate = 2 arrivals per hour. Introduce arrival times,
W0 = 0 and Wk = min [t 0 : N(t) = k] for k 1:
Assume that inspection occurs at t = 5:5 hours.
1. Evaluate conditional expectation of the forth arrival, given that the W10 5:5 < W11
2. Find conditional expectation of the W4, given that the tenth arrival occurred exactly at W10 = 5:5:
Solution
11

## Statistics homework help

For this three-part assessment you will create a histogram or bar graph for a data set, perform assumption and correlation tests, and interpret your graphic and test results in a 2-to-3 page paper.

In this unit we focus on whether two or more groups have important differences on a single variable of interest. For example, for the dependent variable stress score, we may want to know if there is a difference in stress between males and females, or maybe we would like to know if there is a difference in stress levels between people who drink chamomile tea and those who do not, or maybe we would like to determine if a group of expectant parents is less anxious (this is the dependent variable) about the birthing experience after a series of discussions with experienced parents. In each of these examples we have two groups (two groups being compared or the same group being compared before and after), and one dependent variable that is being compared in each group. In this unit you will begin exploring popular statistical techniques (and their assumptions) that are used to compare two or more groups.

The independent t-test, also called unpaired t-test, is typically used in health care to compare two groups of individuals that are entirely unrelated to each other (that is, independent), thus the one group cannot influence the other group. For example, we may wish to compare a drug treatment group to a control group (those not receiving drug treatment) for a specific clinical characteristic (dependent variable) that can be measured at the interval or ratio level (such as cholesterol, depression scale, or memory test).

The dependent t-test, also called paired t-test, compares two groups for a dependent variable measured at the interval or ratio level as well; however, these two groups are in reality just one group. But because they are measured before and after an intervention, we consider them as two groups for analytical purposes. This group is considered dependent because nothing is expected to vary in the nature of the individuals being measured except as a result of the intervention, as the group is composed of the same individuals.

#### Overview

One of the most important steps along the researcher’s path to data analysis is to become familiar with the character of the raw data collected for the project. Before weaving the strands of data into an analytical story that is related to a study’s goals, researchers typically inspect the completeness and quality of the data with various visualization techniques (graphics), summary tables, and mathematical tests of quality (assumption tests), as discussed in Assessment 2. One of these latter tests is a correlation analysis. With this approach, the researcher performs a very basic series of exploratory tests on variable pairs to identify any potentially interesting (yet unknown) relationships between groups of data (variables). Correlational analyses are often later performed as part of the predetermined data analysis plan to answer a specific research question.

#### Demonstration of Proficiency

By successfully completing this assessment you will address the following scoring guide criteria, which align to the indicated course competencies.

• Competency 1: Describe underlying concepts and reasoning related to the collection and evaluation of quantitative data in health care research.
• Interpret the overall clinical meaning and limitations of the relationship of two variables, based on a correlation analysis and literature regarding age and stress.
• Competency 2: Apply appropriate statistical methods using common software tools in the collection and evaluation of health care data.
• Create a histogram and scatter plot for variables tested for normal distribution.
• Perform a normal distribution assumption test for two variables to determine if data is normally distributed.
• Perform an appropriate correlation test to determine the direction and strength or magnitude of the relationship between two variables.
• Competency 3: Interpret the results and practical significance of statistical health care data analyses.
• Interpret the effect size for correlation analysis results.
• Competency 5: Address assignment purpose in a well-organized text, incorporating appropriate evidence and tone in grammatically sound sentences.
• Articulate meaning relevant to the main topic, scope, and purpose of the prompt.
• Apply APA formatting to in-text citations and references.

#### Instructions

For this three-part assessment, complete the following, referring to Yoga Stress (PSS) Study Data Set [XLSX], which you have used previously, as needed.

##### Software

The following statistical analysis software is required to complete your assessments in this course:

• IBM SPSS Statistics Standard or Premium GradPack, version 22 or higher, for PC or Mac.

Please refer to the Statistical Software page on Campus for general information on SPSS software, including the most recent version made available to Capella learners.

##### Part 1: Graphic Representation of the Data from the Yoga Stress (PSS) Study Data Set
1. Create a histogram or bar graph (according to the measurement level of the data) of the following variables: Age, Education, Pre-intervention Psychological Stress Score (PSS).
2. Create a scatter plot of the following pair of variables: Age versus Pre-intervention Psychological Stress Score (PSS).
• Refer to the following resources, as needed, while creating your scatterplot:
##### Part 2: Statistical Tests
1. Perform a preanalysis assumption test for a normal distribution test to determine if the data you intend to use for the correlation tests passes the assumption of being normally distributed.
• You will use this test for Age and Pre-intervention Psychological Stress Score (PSS).
2. Perform the appropriate correlation test to determine the direction and strength or magnitude of the relationship between these two variables from Step 1.
• Remember, we are not concerned about causation at this point and want to determine only if there is a statistical association.
##### Part 3: Yoga Stress (PSS) Study Paper
• Include the histogram and scatter plot graphics you created earlier for Age and Pre-intervention Psychological Stress Score (PSS).
• Provide an interpretation for these graphics.
• Report the statistical outcome of the correlation analysis using appropriate scholarly style, including a brief interpretation of the effect size of the correlation.
• Interpret the practical, real-world meaning (and limitations of the interpretation) of the relationship of these two variables based on the correlation analysis you performed.
• Include the SPSS “.sav” output file that shows your programming and results from Parts 1 and 2 for this assessment.
• Provide at least one evidence-based scholarly or peer-reviewed article that supports your interpretation.

• Length: Your paper will be 2–3 double-spaced pages of content plus title and reference pages.
• Font: Times New Roman, 12 points.
• APA Format: Your title and reference pages must conform to APA format and style guidelines. See the APA Module for more information. The body of your paper does not need to conform to APA guidelines. Do make sure that it is clear, persuasive, organized, and well written, without grammatical, punctuation, or spelling errors. You also must cite your sources according to APA guidelines.

For this three-part assessment you will create a histogram or bar graph for a data set, perform assumption and correlation tests, and interpret your graphic and test results in a 2-to-3 page paper.

In this unit we focus on whether two or more groups have important differences on a single variable of interest. For example, for the dependent variable stress score, we may want to know if there is a difference in stress between males and females, or maybe we would like to know if there is a difference in stress levels between people who drink chamomile tea and those who do not, or maybe we would like to determine if a group of expectant parents is less anxious (this is the dependent variable) about the birthing experience after a series of discussions with experienced parents. In each of these examples we have two groups (two groups being compared or the same group being compared before and after), and one dependent variable that is being compared in each group. In this unit you will begin exploring popular statistical techniques (and their assumptions) that are used to compare two or more groups.

The independent t-test, also called unpaired t-test, is typically used in health care to compare two groups of individuals that are entirely unrelated to each other (that is, independent), thus the one group cannot influence the other group. For example, we may wish to compare a drug treatment group to a control group (those not receiving drug treatment) for a specific clinical characteristic (dependent variable) that can be measured at the interval or ratio level (such as cholesterol, depression scale, or memory test).

The dependent t-test, also called paired t-test, compares two groups for a dependent variable measured at the interval or ratio level as well; however, these two groups are in reality just one group. But because they are measured before and after an intervention, we consider them as two groups for analytical purposes. This group is considered dependent because nothing is expected to vary in the nature of the individuals being measured except as a result of the intervention, as the group is composed of the same individuals.

#### Overview

One of the most important steps along the researcher’s path to data analysis is to become familiar with the character of the raw data collected for the project. Before weaving the strands of data into an analytical story that is related to a study’s goals, researchers typically inspect the completeness and quality of the data with various visualization techniques (graphics), summary tables, and mathematical tests of quality (assumption tests), as discussed in Assessment 2. One of these latter tests is a correlation analysis. With this approach, the researcher performs a very basic series of exploratory tests on variable pairs to identify any potentially interesting (yet unknown) relationships between groups of data (variables). Correlational analyses are often later performed as part of the predetermined data analysis plan to answer a specific research question.

#### Demonstration of Proficiency

By successfully completing this assessment you will address the following scoring guide criteria, which align to the indicated course competencies.

• Competency 1: Describe underlying concepts and reasoning related to the collection and evaluation of quantitative data in health care research.
• Interpret the overall clinical meaning and limitations of the relationship of two variables, based on a correlation analysis and literature regarding age and stress.
• Competency 2: Apply appropriate statistical methods using common software tools in the collection and evaluation of health care data.
• Create a histogram and scatter plot for variables tested for normal distribution.
• Perform a normal distribution assumption test for two variables to determine if data is normally distributed.
• Perform an appropriate correlation test to determine the direction and strength or magnitude of the relationship between two variables.
• Competency 3: Interpret the results and practical significance of statistical health care data analyses.
• Interpret the effect size for correlation analysis results.
• Competency 5: Address assignment purpose in a well-organized text, incorporating appropriate evidence and tone in grammatically sound sentences.
• Articulate meaning relevant to the main topic, scope, and purpose of the prompt.
• Apply APA formatting to in-text citations and references.

#### Instructions

For this three-part assessment, complete the following, referring to Yoga Stress (PSS) Study Data Set [XLSX], which you have used previously, as needed.

##### Software

The following statistical analysis software is required to complete your assessments in this course:

• IBM SPSS Statistics Standard or Premium GradPack, version 22 or higher, for PC or Mac.

Please refer to the Statistical Software page on Campus for general information on SPSS software, including the most recent version made available to Capella learners.

##### Part 1: Graphic Representation of the Data from the Yoga Stress (PSS) Study Data Set
1. Create a histogram or bar graph (according to the measurement level of the data) of the following variables: Age, Education, Pre-intervention Psychological Stress Score (PSS).
2. Create a scatter plot of the following pair of variables: Age versus Pre-intervention Psychological Stress Score (PSS).
• Refer to the following resources, as needed, while creating your scatterplot:
##### Part 2: Statistical Tests
1. Perform a preanalysis assumption test for a normal distribution test to determine if the data you intend to use for the correlation tests passes the assumption of being normally distributed.
• You will use this test for Age and Pre-intervention Psychological Stress Score (PSS).
2. Perform the appropriate correlation test to determine the direction and strength or magnitude of the relationship between these two variables from Step 1.
• Remember, we are not concerned about causation at this point and want to determine only if there is a statistical association.
##### Part 3: Yoga Stress (PSS) Study Paper
• Include the histogram and scatter plot graphics you created earlier for Age and Pre-intervention Psychological Stress Score (PSS).
• Provide an interpretation for these graphics.
• Report the statistical outcome of the correlation analysis using appropriate scholarly style, including a brief interpretation of the effect size of the correlation.
• Interpret the practical, real-world meaning (and limitations of the interpretation) of the relationship of these two variables based on the correlation analysis you performed.
• Include the SPSS “.sav” output file that shows your programming and results from Parts 1 and 2 for this assessment.
• Provide at least one evidence-based scholarly or peer-reviewed article that supports your interpretation.

• Length: Your paper will be 2–3 double-spaced pages of content plus title and reference pages.
• Font: Times New Roman, 12 points.
• APA Format: Your title and reference pages must conform to APA format and style guidelines. See the APA Module for more information. The body of your paper does not need to conform to APA guidelines. Do make sure that it is clear, persuasive, organized, and well written, without grammatical, punctuation, or spelling errors. You also must cite your sources according to APA guidelines.

## Statistics homework help

All posts must 100% original work. NO PLAGIARISM. Post results must be provided using the Excel attached. Make sure you interpret your results on a Word Document.

Using the data set you collected in Week 1, excluding the super car outlier, you should have calculated the mean and standard deviation during Week 2 for price data. Along with finding a p and q from Week 3 excel. Using this information, calculate two 95% confidence intervals. For the first interval you need to calculate a T-confidence interval for the sample population. You have the mean, standard deviation and the sample size, all you have left to find is the T-critical value and you can calculate the interval. For the second interval calculate a proportion confidence interval using the proportion of the number of cars that fall below the average. You have the p, q, and n, all that is left is calculating a Z-critical value,

Make sure you include these values in your post, so your fellow classmates can use them to calculate their own confidence intervals. Once you calculate the confidence intervals you will need to interpret your interval and explain what this means in words.

Do the confidence intervals surprise you, knowing what you have learned about confidence intervals, proportions and normal distribution? Please the Week 5 Confidence T-Interval Mean and Unknown SD PDF and the Week 5 Confidence Interval Proportions PDF at the bottom of the discussion. This will give you a step by step example on how to help you calculate this using Excel.

## Statistics homework help

Provide (2) 150 words response for RESPONSES 1 AND 2 below. Responses may include direct questions. In your peer posts, compare the probabilities that you found with those of your classmates. Were they higher/lower and why? In your responses, refer to the specific data from your classmates’ posts. Make sure you include your data set in your initial post as well. Attached are the excel docs for both responses to help with the post.

RESPONSE 1:

This week we worked with averages, standard deviations, and especialy probabilities.

The first step was to calculate a new standard deviation for a sample size of 4. In my case 14264/SQRT(4).

With this number and the previously calculated mean price of a vehicle, in my example 28232, we first calculate the percent chance that the next four vehicles will be 500 dollars below the mean. I came to a 47.2% chance.

The next probability we calculate is the odds of the next four cars being 1000 dollars above the mean. I came to a 44.4% chance this would be the case.

After that we calculate the odds that the next four cars would cost the same as my mean price. I came to a 50% chance that that would happen.

The final probability to calculate was if the next four cars would cost within 1500 dollars +- of the mean price. I came to a 16.7% chance that would happen.

Speaking for myself I found this post quite challenging and would welcome any critical eyes on work.

RESPONSE 2:

For this week’s forum we are asked to find the normal distribution of a set of vehicles and different probabilities.

The mean of all my vehicles without the supercar is 18,478 and I have a new standard deviation of 685.3123781.

The first question asks for the probability that the price will be less than \$500 dollars below the mean. To figure this out we take my mean and subtract 500 dollars to get 17978. That is p(x<17978). In excel make sure to use norm.dist with a formula of TRUE.

Secondly we are asked to find the probability that the price will be higher than \$1000 dollars above the mean. To figure this out we take the mean and add 1000 dollars to get 19478. That is p(x>19478). In excel make sure to use norm.dist with a formula of TRUE.

Next we are asked to find the probability that the price will be equal to the mean. To figure this out we take the mean and equal it out against itself at 18478. That is p(x=18478). In excel make sure to use norm.dist with a formula of FALSE.

Finally we are asked to find the probability that the price will be \$1500 within the mean. To figure this out we take the mean 18478, and subtract 1500 as well as add 1500 in a separate equation. Within excel I input the normal distribution formula with the inputed numbers of =NORM.DIST(19978,16978,C28,TRUE)-NORM.DIST(16978,18478,C28,TRUE).

## Statistics homework help

Write a 2- to 3-page critique of the research you found in the Walden Library that includes responses to the following prompts:

• Why did the authors select binary logistic regression in the research?
• Do you think this test was the most appropriate choice? Why or why not?
• Did the authors display the results in a figure or table?
• Does the results table stand alone? In other words, are you able to interpret the study from it? Why or why not?

## Statistics homework help

Dr. Beeper, an Educational Psychologists who studies issues related to higher education, is interested in studying key factors that impact year to year persistence among college students.  His review of the literature identifies several factors that appear to be causally related to persistence. Specifically, academic aptitude, goal commitment, institutional commitment, and the number of work hours.

To test the importance of these factors, Dr. Beeper administers a set of questionnaires to 100 randomly selected first-time, full-time freshmen college students (50 male and 50 female) that attended the Freshmen Orientation in the Fall of 2016, at Newton Young University (NYU) in Nebraska.

Measures:

Institutional Commitment (IC) represents the importance that students place on graduating from the college they are currently attending.   Institutional Commitment was measured with five-item questionnaire. Each item was rated on a 0, 1, or 2 scale.  The possible range of scale scores are zero to 10, where values close to zero indicate little to no importance, and values close to 10 indicate high importance.

Goal Commitment (GC) represents the importance that students place on obtaining a college degree.  Goal Commitment was also measured with five-item questionnaire.  Each item was rated on a 0, 1, or 2 scale.  The possible range of scale scores are zero to 10; where values close to zero indicated little to no importance to obtaining a college degree, and values close to 10 indicated a high importance to graduating from college.

Academic Aptitude was represented as scores on both the SAT-Math and the SAT-Verbal tests.  SAT scores for all participants were obtained from high school transcripts.

Hours works, represents the anticipated number of hours the student expected to work throughout the semester.

Finally, Year-to-year persistence was determined by examining the enrollment records for the sample of 100 students. A student that was registered for registered for the Fall 2017 classes was classified as a “Persister”, and given a code of 1, a student that did not re-enroll for classes at NYU, or any other college/university (based on follow-up phone interviews) was considered a “Non-persister”, and was given a code of 0.  Therefore, the SPSS variable Persist has two levels, 0 and 1.

The assignment is, using the attached SPSS data file, conduct a binary logistical regression analysis in which IC, GC, SAT-MathSAT-Verbal, and Hours Worked are the predictor. variables (covariates in SPSS), and the variable Persist is the outcome (DV in SPSS). Use my sample summary as a model for your summary.

The specific elements of the assignment are:

1) Create a Null and Alternative Hypotheses for the Logistical Regression Analysis

2) State the Goals of the analysis

3) Summarize the results and interpret findings the overall model (for example the Chi Square results, Nagelkerke R-Square or Cox Snell R-Square).

4) Summarize and interpret the results for each predictor;  and present, summarize and interpret the results for each significant predictor (i.e., B, Wald’s test, df, p and OR (ExpB). Interpret the significant OR using the effect size conventions I posted in last week’s (8) discussion board.

5) Include and refer to the appropriate tables within the summary.

Please read my sample summary see what statistics to report, and how to report and interpret them in correct APA style, as well as the tables to include.

You’ll see that in my sample summary I also include t-tests. You may  want to conduct t-tests that compare “persisters” and non-persisters, on the predictor variables (covariates).  Please note that the t-test are optional, and will have no impact on your grade whether you include them or not.  The t-test  are very informative about the bivariate relationship between the predictor variables (covariates in SPSS) and the binomial outcome (DV in SPSS) .

Please note that you are not required to conduct the t-tests, or to compute and report Cohen’s d.

Here’s the syntax for my sample summary.

T-TEST GROUPS=BO(0 1)
/MISSING=ANALYSIS
/VARIABLES=teachsat ressat wkoverld
/CRITERIA=CI(.95).

LOGISTIC REGRESSION VARIABLES BO
/METHOD=ENTER teachsat ressat wkoverld
/PRINT=GOODFIT CI(95)
/CRITERIA=PIN(0.05) POUT(0.10) ITERATE(20) CUT(0.5).