Statistics
Q1. [10 points] Consider that a researcher conducted a study that examined resilience attitudes of 100 women, who participated in an educational intervention over time. The variable “resilience was measured by a Likert scale with 30 items statements; each scaled on 1 to 5 points. The point of 1‚ meant the lowest level of resilience, while the point of 5‚ meant the highest resilience. A higher total score was interpreted as “higher level of resilience. The researcher first measured the resilience of these women before they entered the educational intervention as a pre-test, and found that at pre-test, the resilience scores were normally distributed with a Mean = 85 and Standard Deviation (SD) = 2.5. [Hint please keep in mind the rules on “normal distribution when figuring out the questions, and you may want to review slides on hypothesis testing as well] Answer the following set of questions based on rules governing the normal distribution of scores.
(a) What was the median resilience score for these women at pre-test, and why? Answer: 85 because median = mean in a normal distribution
(b) What was the modal resilience score for these women at pre-test, and why? Answer: 85 because mode = mean in a normal distribution
(c) What were the two scores of resilience between which 95% of women scored at this pre-test? Please explain/show how you calculated/derived these two scores: Answer: 95%, or .95 x 100 = 95 adults
(d) Consider that “Anna is one such participating woman, and Anna scored 1SD below the mean at pre-test. Please indicate Annas score on resilience, and explain/show how you calculated/derived her score: Answer: 68% of adults, or .68 x 100 = 68 adults (d) Consider that “Shakila is another woman, whose reliance scores were collected at pre-test, and she actually scored 1SD above the mean in her pre-test resilience. What percent of participating women had resilience scores between Anna and Shakila, and briefly explain how you derived your answer: Answer: (e) Consider that Anna scored 1 SD below the mean in her resilience, and Shakila scored 1 SD above the mean in her resilience. At what percentile for resilience did each woman score on the pre-test? Explain and show how you calculated/derived your answer. Answer: **********************
Q2. [10 points] Now lets stay with the research study example described in Q1. Consider then that at pre-test, the Pre-Mean = 85 with SD = 2.5, and that the pre-test scores were normally distributed. Later, when the researcher conducted a post-test for the womens resilience scores, and the Post- Mean = 90 with SD = 5.5, but the scores at post-test were no longer normally distributed, in fact, post-test scores were positively skewed. Answer the questions below.
(a) In which distribution of resilience scores – the pre-test or the post-test would you find a median resilience score that was lower than the average resilience score, and why? Answer:
(b) In which distribution of resilience scores – the pre-test or the post-test would you find the scores more heterogeneous, and why? Answer: 1
(c) Now, consider a slightly different situation. Imagine that you are looking at a printout of scores for a different research problem, and that you are looking at 2 sets of scores, and that each set of scores, gives you the scores for mode, median, and the mean (respectively in that order). All you know that one set is supposed to be a pre-test with normally distributed scores and another set is supposed to be a post-test with negatively skewed scores. The SET A shows 86, 85, 80 for mode, median, and mean. The SET B shows 90, 91, 91 for mode, median, and mean. Explain which set of scores SET A or SET B should be attributed to pre-test normally distributed scores, and which set to post-test negatively skewed scores, and why: Answer: Set B should be attributed to pre-test normally distributed scores The SET A set to post-test negatively skewed scores: mean is being deflated and so it will be smaller than the median or the mode ().
(d) Now, consider that the researcher formulated the following research question: For women who struggle with child neglect, is there a statistically significant correlation between pre-test and post-test scores on resilience? Please state the “null hypothesis for this study: Answer: Variables of women who struggle with child neglect will not significantly predict the correlation between pre-test and post-test scores on resilience
(e) Now, let’s assume that the researcher tested the null hypothesis for the correlation between pre-test and post-test scores on resilience, and found that the Pearson correlation result r = 0.13 at p = 0.24. Examine this result, and indicate which type of statistical error (type 1 or type 2) could the researcher commit in interpreting these results, and why: Answer: ************************
Q3. The following question is going to focus on your knowledge of z-scores, and their specific use in figuring out percentiles. For this question, you need to be able to use the zscore table in Appendix B in the Salkind textbook, and apply the zscore formula (Score – Mean / SD = zscore) in figuring out the percentile position of scores. You may also want to review the slides on hypothesis testing, and the handout problems, which show you the steps. Hint: You must always remember important rules in translating zscores to percentile position:
(a) Mean score = z = 0, but that position occurs at 50th percentile; always!,
(b) If z-score is positive , then that positive zscore occurs at a percentile above the mean score (at a percentile that is higher than the mean score!).
(c) If z-score is negative, then that negative zscore occurs at a percentile below the mean score
(d) For example, you should be able to figure out that a score 1 SD above the mean corresponds 1z, and that corresponds to 34.13% area in the table between the mean and the zscore of 1, and therefore, the score will be found at 84.13th percentile. Now, let’s consider that a researcher studies the “length of homelessness†of adult individuals. Shelter X will take individuals who are homeless, and based on their statistics that they collect for the length of homelessness prior to entering into the shelter, the homelessness Mean = 50 days, SD = 10.5 days for individuals, who do well in the shelter. The shelter director notes that those individuals who come to the shelter with “their prior length of homelessness that is greater (>) than 3 SD from the shelter “average†length of prior homelessness, typically do not do well, and need many other kinds of support services that the shelter is not currently able to provide. Answer questions below: (a) At what percentile of prior length of homelessness is“Tarik who reports that he has been homeless for 85 days? Explain and show how you derived/calculated your answer: [Hint: 85 50 / 10.5 = _zscore] Answer: 85 50 / 10.5= 3.33 (b) At what percentile of prior length of homelessness is “Omar who reports that he has been homeless for 63 days? Explain and show how you derived/calculated your answer: Answer: 63-50/10.5=1.24 (c) Now, consider both Tariks and Omars prior homelessness, and explain, which of these two men is more likely to benefit by shelter services and which one is not, and why? Answer: *************************
Q4. Now, lets focus on the use of one sample Ztest (chapter 10 in Salkind book) and its interpretation in hypothesis testing. Please review slides as well. Please remember that the Ztest test can be used when we have a sample mean score and sample size, and we want to compare the sample average (mean) with a known population mean score, given that we also know population standard deviation. Now, please focus on the following case scenario information, which focuses on the number of flu cases in the state versus a local school:
Case Scenario: In the state X, researchers have uncovered that for the entire state population, the population Mean = 16.00 flu cases per week, and the population standard deviation = 15.10 flu cases. These researcher then examined the flu cases during this past season in the sampled city at one location within this state, and found that for a sample of N = 500 cases, the sample Mean = 15 flu cases per week. They asked the following question: Is there a significant difference in the mean number of flu cases between the state population mean number of flu cases and the sampled city mean flu cases? a) State/ write the 2-tail research hypothesis for this study scenario: (b) State/write the null hypothesis for the research question: (c) Now, please remember that you have learned that zscore of + or -1.96z corresponds to p = 0.05, and these z-score values create boundaries for 95% of scores that fall within 2 SD around the mean “by chance. And, that these two z-score values are the “critical zscores that are used for making decision regarding statistical significance for “calculated zscore in the Ztest results. Before you tackle this problem, please review how to make an interpretation between “calculated zscore and “critical zscore in order to figure out statistical significance. This problem scenario is reviewed in the Salkind text chapter and the provided slides! So, now calculate the Ztest given the case scenario for the flue cases in which you compare the sample mean with the population mean. Dont forget to calculate the SEM correctly as well and show your work! Show your work and the result here: SEM = Population standard deviation / Square Root of Sample N Ztest = Sample Mean Population Mean / SEM = zscore(calculated) (d) Now, please compare your calculated z-score result (from the Ztest) with the critical zscore of + – 1.96z, and draw conclusion about statistical significance for this comparison. Explain whether the sample of 500 kids in the city X public school system was significantly sicker than or as sick as kids in the state population: (e) Based on your results, what type of statistical error (Type 1 or Type 2) could have been committed in accepting these results? **********************
Q5. [10 points] Now, lets see what you have learned about interpreting the direction in a significant correlation between two variables. Please interpret the direction (positive, negative, non-linear) in the following statements on the relationship of two variables. (a) A researcher found that more days being absent from school significantly correlated with a lower score on a comprehensive exam test. The variable direction is: (b) A researcher found that male (code = 0) students were significantly more days absent from class than female (coded = 1) students, who were fewer days absent from class. The variable direction is: (c) A researcher also found that male (code = 0) students had significantly fewer number of difficulties in a group discussion, when compared with female (coded = 1) students, who had more number of difficulties in a group discussion. The variable direction is: (d) A researcher found that lower scores in self-esteem significantly correlated with lower levels of anxiety scores. The variable direction is: (e) A researcher found that lower scores on satisfaction with supervision in field internships significantly correlated with higher levels of distress. The variable direction is: ***********************
Q6. [10 points] Please keep focusing on the Pear correlation test r. In the provided results below, you are asked to interpret the results of these findings. Please pay attention to what the question is asking you! (a) A researcher found that there is a significant correlation between the number of days absent from school and the scores on a reading test: r = – 0.45, p < 0.05. Interpret only the direction of the relationship between the variables in a meaningful English language: (b) A researcher found that there is a significant correlation between the level of anxiety (measured by a score on an Anxiety Scale) and the number of conflicts students have at school during the day: r = 0.65, p < 0.05. Interpret the magnitude and the direction of the relationship between the variables in a meaningful English language: (c) A researcher found that there is a correlation between the level of anxiety (measured by a total score on an Anxiety Scale) and the number of conflicts students have at school during the day: r = 0.65 at p = .031. Interpret whether this result was statistically significant, and explain why or why not: (d) A researcher found that there is a correlation between the level of anxiety (measured by a score on an Anxiety Scale) and students’ gender (Male = 0, Female = 1): r = 0.11 at p = 0.321. Interpret the statistical significance and the meaning of this correlation result: (e) A researcher found that there is a correlation between the level of quality communication of teens (measured by a total score on a Communication Scale) and the level of parental empathy (measured by a total score on an Empathy Scale): r = 0.35 at p = 0.001. Interpret the statistical significance and provide the full meaning of this correlation relationship: **********************
Q7. [10 points] Now, please focus on the concept of “regression for predicting relationship between variables. You may want to review understanding MRA tables provided and discussed in the slides! Consider that a researcher wanted to investigate whether variables of age, gender, grades during senior year in high school, and engaging in extra-curricular activity would significantly predict GPA of the first year college students. Please pay attention to the information in the table and under the table as well! Examine the table below and answer questions below the table: Table X. Prediction of GPA by Age, Gender, Grade, and Engagement in Extracurricular Activities of Randomly Selected College Students (N = 218) Unstandardized Coefficients Standardized Coefficients Zero-ordera r B St. Error (SE) Beta t Sig. p-value (Constant) 0.353 .11 .345 .544 Age .11 0.012 .13 .121 .435 .322 Gender .45* 0.495 .43 .315 3.95 .031 Grade in HS .75** 1.350 .22 .569 4.22 .001 Extra-Activity .48** 0.651 .18 .355 4.01 .001 Model Adjusted R-square = 0.55 Gender: 1 = Female, 0 = Male a Simple bivariate Pearson r correlation * p < 0.05, ** p < 0.001 (a) State/write the null hypothesis for this study using non-directional (2-tail) approach: (b) Identify the dependent variable in this study: (c) Examining the table, which predictor variables significantly predicted the dependent variable? Identify these variables, and explain how you know: (d) Examine the result for gender, and explain the direction of this relationship between this variable and the dependent variable: (e) Explain the percent of variance in the dependent variable that is accounted for by this model with the four predictor variables: *****************
Q8. [10 points] Please keep focusing on regression and prediction of outcomes. You will need to engage in “internet google based search and research in order to complete this question. Case scenario: Consider that a researcher wants to study which variables may help to predict symptom severity in functioning of elderly individuals with Alzheimers disease. Based on a literature review, this researcher is considering to use the variable of the level of education and the variable of the general physical health (rated on 1 to 10 scale) as two starting variables. This researcher is also measuring the severity symptoms in human functioning in the Alzheimers disease by a numeric scale called FAST (which measures human functioning along a scale from 1-7, and higher the stage the more pronounced the symptoms of Alzheimer disease). Answer the following questions:
(a) Please engage and do a google scholar search, and identify 2 other potentially important variables predictors and please provide the APA-6 reference for the studies from which you derived the 2 other potential predictors) that could be studied in this context:
(b) Please explain why, or on what logical basis, or evidence, do you think it would be important for social workers to include these two added variables in a study addressing the symptom severity in functioning of elderly individuals with Alzheimers disease.
(c) With the four predictor variables level of education, general physical health, and the two variables that you identified – Please state/write the research question that could be used to conduct a multiple regression analysis (MRA): **********************************
Q9. [10 points] Now, please find a study that has used Multiple Regression Analysis (MRA) on any topic relevant for social workers, and 1. Provide an APA-6 reference for this study 2. Briefly explain in a paragraph what the overall study was about: 3. Explain how researchers used MRA for analysis in this study, and give a brief (one paragraph) explanation on what they found specifically about the MRA results. **********************************
Q10. [10 points] Final question is asking you to focus on the chi-square test, and the frequency association between variables. (a) Consider that a researcher investigated whether is a significant association between depression (measured as low and high) and caregivers burden in family members who provide care to a member with Alzheimers disease (measured as low, moderate, and high burden). In this study, the researcher was particularly interested if different levels of caregiver burden had different consequence (association) for different level of depression. Examine the statistical results below and then answer questions posed below the tables. DEPRESSION * BURDEN Crosstabulation BURDEN Total Low Moderate High DEPRESSION LOW Count 79 56 35 170 % within Burden Levels 78.2% 57.7% 34.3% 56.7% HIGH Count 22 41 67 130 % within Burden Levels 21.8% 42.3% 65.7% 43.3% Total Count 101 97 102 300 % within Burden Levels 100.0% 100.0% 100.0% 100.0% Chi-Square Tests Value df Asymp. Sig. (2-sided) Pearson Chi-Square 39.903a 2 .000 Likelihood Ratio 41.330 2 .000 Linear-by-Linear Association 39.713 1 .000 N of Valid Cases 300 a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 42.03. Symmetric Measures Value Approx. Sig. Nominal by Nominal Phi .365 .000 Cramer’s V .365 .000 N of Valid Cases 300
(a) State/write the null hypothesis for this study:
(b) Interpret the Pearson chi-square result and what the findings indicate about the significant association between depression and caregivers burden:
(c) Explain the magnitude or effect size in the result:
(d) Lastly, examine the percentages of “burden groups in columns, and explain which level of burden was associated with what kind of depression:
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