The Descriptive Data for Ethnicity/Race Group (Math Problem Sample)

Instructions: For each question, answer all items. Make sure that you are detailed when you answer the questions. As a rule of thumb, when you are answering any questions using statistics, you should include all statistics in the answer (e.g. if it asks about the mean, put the actual value of the mean in the sentence). Type your answers out underneath each question either bolded or in a different font color (or both). Please do not delete the questions or the output. When you are finished with the assignment, upload it as a .doc, .docx, or .pdf to the assignment on Canvas.
The analyses come from the high school exam data set. This data set includes information about how students in NY State performed on a bunch of standardized tests. In addition to the test scores, there is demographic information (ethnicity/race, gender) and school information (type of school, type of program). The analyses below use a variety of tests that look at the relationships between variables.
Note: There are 5 parts to this assignment each one on a separate page. Make sure to answer all parts.
Part I: One-Way ANOVA
This analysis is looking at the relationship between ethnicity/race and freshman year science scores. Use the tables and graph included below to answer the following questions:
1 What are the null and alternative hypotheses for this analysis?
Null hypothesis: there is no significant difference in the mean freshman year science scores among the different ethnic groups
Alternative hypothesis: there is a significant difference in the mean freshman year science scores among the different ethnic groups
2 In a few sentences, describe the descriptive data for each ethnicity/race group (include N, mean, standard deviation, minimum, & maximum).
In a sample of 24 Hispanics, the mean freshman year science scores is 45.38 points with a standard deviation of 8.22 points. The ethic group has a minimum score of 26 points and a high score of 63 points. On the other hand in a sample of 11 Asians, the mean freshman year science scores is 51.45 points with a standard deviation of 9.5 points. The group has a minimum score of 34 points and a maximum of 66 points. In addition, in a sample of 20 African-amer, the mean freshman year science scores is 42.8 points with a standard deviation of 9.44 points. The minimum score for the ethnic group is 29 points with a maximum of 61 points. As for the whites, in a sample of 145, the mean freshman year science scores is 54.2 points with a deviation of 9.1 points. The maximum score is 74 points and the minimum is 33 points.
3 Did we violate the assumption of homogeneity in this analysis? Include evidence for your answer (i.e. how do you know?).
As presented in the Levene statistic test for homoscedasticity, the statistic is 0.565, with a p-value = 0.639. The null hypothesis for the test is that homoscedasticity is true (constant variance), and the alternative is that there is heteroscedasticty (no constant variance). Undertaking the test at 95% confidence level, then we are saying that the level of significance is 0.05. The decision rule is to reject the null hypothesis if the determined p-value is less than the level of significance. In our case 0.639>0.05, thus we fail to reject the null hypothesis. Therefore, the assumption of homoscedasticity is not violated.
4 Were there significant differences in science scores between the different ethnicities?
To assess the hypothesis of equal mean among the groups, verses unequal means in the groups we make use of the ANOVA tests since the groups are more than two. The results as indicated on the ANOVA table show that the f-value = 14.021, with a p-value=0.000. Undertaking the test at 95% confidence level, then we are saying that the level of significance is 0.05. The decision rule is to reject the null hypothesis if the determined p-value is less than the level of significance. In our case 0.000<0.05, thus we reject the null hypothesis. Therefore, we conclude that there is significant difference in the mean freshman year science scores among the different ethnic groups.
5 Based on the results of your analysis, should you analyze the post hoc tests? yes
1 If yes, which groups are significantly different from each other? (List all).
Based on the post Hoc test of multiple comparisons, the groups that have statistically significant difference in the mean freshman year science scores include; Hispanic and Asian that have a mean difference of 6.08 points which is significant at 10%. The other groups that are significantly different is Hispanic and the whites with a mean difference of 8.83 points which is significant at 1% level. The other is the Asian and African-amer with a difference of 8.65 points that is significant at 5% level. The other is the African-amer and the whites with a mean difference of 11.4 points which is significant at 1% level. Thus at 5% level of significance the groups with statistically different means include:
* Hispanic and the whites
* Asian and African-amer
* African-amer and the whites
2 Explain what the post hoc tests tell us about the relationship between ethnicity and science scores.
The post hoc test is an in-depth analysis that tries to give more meaning to the results of the ANOVA test. The anova although will tell us that the means are not equal, it will not show which specific mean score not equal, which is explained in the post hoc test, that will bring out the actual groups that have statistically significant difference in their mean score. From this results it’s evident that the science score is influenced by ethnic groups. And not all relations but the science score will be different if we have Hispanics and whites, African-amer and whites/Asians. Other ethnic relations will not have an effect on the science score.
6 Complete Step 4 of hypothesis testing. Include your decision about the null hypothesis, a sentence describing the results (including post hoc tests), and the APA-style string of statistics.
The mean science score will be significantly rising with change from Hispanic group to Asian ethic groups. However, the mean score will significantly drop with change from Asian ethnic to African-amer group, but will significantly rise if the group is changed to white. Thus the group with the significantly high science score is the whites, and the lowest score is within the African-amer group. Step four of hypothesis testing entails determination of the test statistic as well as the associated p-value to decide if results are statistically significant. The null hypothesis is that the means are the same for the different combinations.
The test statistic is given by the t=ẋ1-ẋ2-(µ1-µ2)s.e of the diff.. The relationship between hispanics and Asian has a tets statistic of 1.844 (6.079/3.296), with a p-value of 0.067 which is more than 0.05 level and thus the results are not statistically significant. Hence the mean science score between the Hispanic and Asians is the same. The relationship between Hispanics and African-amer has a test statistic of 0.9395 (2.575/2.74069) with a p-value =0.349 which is more than 0.05. Thus the mean science score between Hispanic and the African-amer is the same. The relationship between the Hispanic and white has a test statistic of 4.424 (8.825/1.99484), with a p-value of 0.000 which is less than 0.05. This means that the mean science score between the Hispanics and the whites is statistically different from each other. The relationship between Asian and African-amer has a test statistic of 2.547 (8.65455/3.39801) and a p-value of 0.012 which is less than 0.05. Hence meaning that the mean science score is statistically different between these two groups. The test statistic between the Asian and whites is 0.9678 (2.74545/2.83098) with a p-value of 0.333 which is more than 0.05, thus the mean for the two groups is the same. The relation between African-amer and whites has a tests statistic of 5.28 (11.4/2.15922), with a p-value of 0.000 which is less than 0.05, thus the mean science score between these two groups is statistically different.
7 In your own words, what is the relationship between ethnicity and science scores?
The results of the study reveal that there is significant difference in the mean science score between the four groups under consideration. Specifically, the mean science score between Hispanic and the whites, Asian and African-amer, and African-amer and the whites are different from each other. Thus the science score will be influence by the ethic group one comes from mainly if we are comparing Hispanic and the whites, Asian and African-amer, and African-amer and the whites, with other relations have no relationship between the ethic group and score since the mean score is the same.
Descriptives

freshman yr science score


N

Mean

Std. Deviation

Std. Error

95% Confidence Interval for Mean

Minimum

Maximum






Lower Bound

Upper Bound



hispanic

24

45.3750

8.21881

1.67766

41.9045

48.8455

26.00

63.00

asian

11

51.4545