Introduction and Statement of Problem Research Paper - 9075 Words

Introduction and Statement of Problem (Research Paper Sample) Content: NameInstructorCourseDateIntroduction and Statement of ProblemIn a study involving handicapped students, researchers attempted to predict GPA from two demographic variables (sex and ethnicity) and two independent measured variables (interview score and contact hours) used in counseling and/or tutoring. A dummy variable for sex was taken as 1 = male, 2 = female. For ethnicity: 1 = African-American, 2 = Hispanic, and 3 = white, non-Hispanic. The data is shown in the table below.DATA TABLEStudents Ethnicity Sex Interview Hours GPA 1 1 2 11.0 4.0 5.50 2 1 2 10.0 5.0 4.10 3 1 2 12.0 73.0 5.00 4 1 2 11.5 68.0 4.22 5 1 2 10.8 82.0 5.00 6 1 1 12.5 72.5 5.00 7 1 1 9.5 64.0 4.60 8 1 1 9.5 78.0 4.25 9 1 1 8.0 64.0 4.00 10 1 1 7.5 13.0 2.00 11 2 2 9.0 37.0 4.25 12 2 2 8.2 4.0 4.00 13 2 2 10.7 38.5 4.61 14 2 2 8.5 3.0 2.93 15 2 2 12.5 10.5 5.50 16 2 1 12.0 80.0 4.77 17 2 1 12.2 6.0 5.00 18 2 1 7.0 6.5 3.25 19 2 1 8.6 22.0 2.66 20 2 1 8.3 28.5 3.37 21 3 2 10.9 12.0 5.00 22 3 2 9. 0 9.0 4.00 23 3 2 10.0 5.0 5.00 24 3 2 7.2 12.0 3.87 25 3 2 8.5 4.0 3.00 26 3 1 10.0 8.0 4.77 27 3 1 8.5 8.0 5.00 28 3 1 10.0 22.0 5.08 29 3 1 11.4 61.5 5.57 30 3 1 11.9 37.0 6.00 We are to do a complete analysis of the data and justify the expression equation that will adequately "fit" the data.Type of ExperimentThis is a general situation in which Y is a function of several independent variables X1, X2, à ¢Ã¢â€š ¬ Xk with no restrictions on the settings of these k independent variables. The X variables have already acted and we simply record their values along with those of the dependent variable Y. This is ex-post-facto research, as exposed to experimental research in which one manipulates that Xà ¢Ã¢â€š ¬s and observes the effect on Y.In this case, we wish to predict the GPA of students. Y will be a function of ethnicity, sex, interview, and hours (X1, X2, X3, and X4 respectively).Mathematical Model for Multiple RegressionThe mathematical model for this particular experimen t is:yà ¢Ã¢â€š ¬ = b0 + b1x1 + b2x2 + b3x3 + b4x4b0 = The y-intercept in the equationb1 = The coefficient for ethnicity type (we will multiply this coefficient by 1, 2, or 3, depending on what the studentà ¢Ã¢â€š ¬s ethnicity is)x1 = The value we can substitute in for ethnicity to predict GPA in the final equation (in this experiment, we are using the numbers 1-3 to substitute for ethnicity type)b2 = The coefficient for sex (we will multiply this coefficient by 1 or 2, depending on what the studentà ¢Ã¢â€š ¬s sex is)x2 = The value we can substitute in for sex to predict GPA in the final equation (in this experiment, we are using the numbers 1 or 2 to substitute for sex)b3 = The coefficient for interview score (we will multiply this coefficient by the studentà ¢Ã¢â€š ¬s interview score)x3 = The value we can substitute in for interview score to predict GPA in the final equationb4 = The coefficient for contact hours (we will multiply this coefficient by the studentà ¢Ã¢â€š ¬s cont act hours)x4 = The value we can substitute in for contact hours to predict GPA in the final equationHypotheses TestingWe need to test nine hypotheses. We need to test whether each of the four factors (ethnicity, sex, interview score, and contact hours) have a significantly different effect on GPA. It would also be good to see if ethnicity and sex significantly affect the interview scores and contact hours. Lastly, we want to see if the interview score has a significant effect on the number of contact hours. The nine sets of hypotheses are below.Ethnicity Hypothesis on GPANull Hypothesis (H0): The means of the three ethnicities have the same effect on GPA.Alternate Hypothesis (HA): One of the ethnicities has a significantly different effect on GPA than the other ethnicities.Sex Hypothesis on GPANull Hypothesis (H0): The means of the two sexes have the same effect on GPA.Alternate Hypothesis (HA): One of the sexes has a significantly different effect on GPA than the other sex.Inte rview Score Hypothesis on GPANull Hypothesis (H0): The means of the twenty-one different interview scores have the same effect on GPA.Alternate Hypothesis (HA): One of the interview scores has a significantly different effect on GPA than the other interview scores.Contact Hours on GPANull Hypothesis (H0): The means of the twenty-two different contact hours have the same effect on GPA.Alternate Hypothesis (HA): One of the numbers of contact hours has a significantly different effect on GPA than the other numbers of contact hours.Ethnicity Hypothesis on Interview ScoreNull Hypothesis (H0): The means of the three ethnicities have the same effect on GPA.Alternate Hypothesis (HA): One of the ethnicities has a significantly different effect on GPA than the other ethnicities.Ethnicity Hypothesis on Contact HoursNull Hypothesis (H0): The means of the three ethnicities have the same effect on the number of contact hours.Alternate Hypothesis (HA): One of the ethnicities has a significantly di fferent effect on the number of contact hours than the other ethnicities.Sex Hypothesis on Interview ScoreNull Hypothesis (H0): The means of the two sexes have the same effect on the interview score.Alternate Hypothesis (HA): One of the ethnicities has a significantly different effect on the interview score than the other ethnicities.Sex Hypothesis on Contact HoursNull Hypothesis (H0): The means of the two sexes have the same effect on the number of contact hours.Alternate Hypothesis (HA): One of the ethnicities has a significantly different effect on the number of contact hours than the other sex.Interview Score on Contact HoursNull Hypothesis (H0): The means of the twenty-one different interview scores have the same effect on the number of contact hours.Alternate Hypothesis (HA): One of the interview scores has a significantly different effect on the number of contact hours than the other sex.Graphical/Descriptive AnalysesIn order to understand the behavior of the data, some graph ical and descriptive analyses will be presented. We are going to look at boxplots for each of the nine hypotheses that were represented above.Boxplot of GPA Versus EthnicityThe boxplot of GPA versus ethnicity below was generated by Minitab.Boxplot InterpretationsThe boxplot above showed the means of the GPAs for the three ethnicities (1 = African-American, 2 = Hispanic, 3 = white, non-Hispanic). A line also joined the three means.If we compute the three means manually, we would see that the mean GPA for African-Americans is:(5.50 + 4.10 + 5.00 + 4.22 + 5.00 + 5.00 + 4.60 + 4.25 + 4.00 + 2.00)/10 = 43.67/10 = 4.367The mean GPA for Hispanics is:(4.25 + 4.00 + 4.61 + 2.93 + 5.50 + 4.77 + 5.00 + 3.25 + 2.66 + 2.37)/10 = 39.38/10 = 3.938The mean GPA for white, non-Hispanics is:(5.00 + 4.00 + 5.00 + 3.87 + 3.00 + 4.77 + 5.00 + 5.08 + 5.57 + 6.00)/10 = 47.29/10 = 4.729By looking at the boxplots, it is obvious that the mean GPA for white, non-Hispanics is the greatest while the mean GPA f or Hispanics is the smallest. This was confirmed when we computed the means manually. All three boxplots overlap one another. The boxplot for African-Americans has an outlier, which is 2.00. The ethnicity seems to be statistically significant for GPA based on our observations. However, we still need to do further analysis in order to support or reject this claim.Boxplot of GPA Versus SexThe boxplot of GPA versus sex below was generated by Minitab.Boxplot InterpretationsThe boxplot above showed the means of the GPAs for the two sexes (1 = male, 2 = female). A line also joined the two means.If we compute the two means manually, we would see that the mean GPA for males is:(5.00 + 4.60 + 4.25 + 4.00 + 2.00 + 4.77 + 5.00 + 3.25 + 2.66 + 3.37 + 4.77 + 5.00 + 5.08 + 5.57 + 6.00)/15 = 65.32/15 = 4.355The mean GPA for females is:(5.50 + 4.10 + 5.00 + 4.22 + 5.00 + 4.25 + 4.00 + 4.61 + 2.93 + 5.50 + 5.00 + 4.00 + 5.00 + 3.87 + 3.00)/15 = 65.98/15 = 4.399By looking at the boxplots, it is obvious that the mean GPA for males is greater than the mean GPA for females. This was confirmed when we computed the means manually. However, the difference is not great at all. The two boxplots overlap each other. There are no outliers present ...