Part 1

Introduction

The population we want to compare are performance in exams of students whose race/ethnicity are grouped B/C/D.

Complete data can download from https://www.kaggle.com/spscientist/students-performance-in-exams. This dataset includes scores from three exams and a variety of personal, social, and economic factors that have interaction effects upon them. There are three quantitative variables and five categorical variables. Group B has 190 records, group C has 329 records, group D has 262 records.商科统计代写

Qualitative variables we observed were “gender” and “parental level of education”. “gender” male means students are male, “gender” female means students are female. “parental level of education” gives the level of education of students’ parents.

Quantitative variables we observed were “math score” / “reading score”/ “writing score” which are relative exam scores.商科统计代写

Part 2

Sample and collect data

We use simple random sampling without replacement to get the subset of the complete dataset.

Step1: add a column as id number series for each group B/C/D.

Step2: use Sampling tool that’s part of the Data Analysis command, set id number column as the input range, then remove the replicate items, make sure the sample size is 30.商科统计代写

Frequency Tables and Pie Charts

For each qualitative variable, make a frequency table and a pie chart.

1) gender

The pie chart above show the gender of the 30 students exam records sampled from race group B/C/D. 商科统计代写we can see that the number of female students are equal with the number of male students in group B samples, the number of female students are less than the number of male students in group C and group D samples.

2) parental level of education

The pie charts above show parental level of education of the 30 samples from race group B/C/D. we can see that student’s parental level of education in group D has the highest proportion of bachelor’s degree. High school proportions are same in group B and group C.

Part 3

Box Plot

The boxplots above to the left shows the math scores of the 30 students sampled from race group B, group C, and group D. 商科统计代写The boxplots show that group C has the least variation in math scores, while group D has the largest variation in math scores. Also, we can see that the medium math score of group D is higher than that of group B and group C.

The boxplots above in the center show the reading scores of the 30 students sampled from race group B, group C and group D. the boxplots show that group B has the largest variation. Once again, the median reading score of group D is higher than that of group B and group C.

The boxplots above in the right show the writing scores of the 30 students sampled from race group B, group C and group D. the boxplots show that the group B has the smallest variation. Once again, the median writing score of group D is higher than that of group B and group C.

Recall that the parental level of education in subgroup C have the highest proportion in bachelor’s degree, maybe students who parents with higher education are more likely to get higher score in their own exams.

Part 4

Inferential Analysis

The table below shows the mean, standard deviation, and the mean’s margin of error for each quantitative variables and population. 商科统计代写The margins of error are calculated using a normal distribution and a 95% confidence level.

	population	groupB	groupC	groupD
math scores	mean	60.77	64.63	67.80
	standard deviation	17.43	14.46	15.79
	margin of error	6.24	5.17	5.65
reading scores	mean	63.87	68.33	69.03
	standard deviation	18.14	15.10	17.18
	margin of error	6.49	5.41	6.15
writing scores	mean	62.07	66.57	68.73
	standard deviation	18.50	17.22	17.00
	margin of error	6.62	6.16	6.08

The charts below show the means and confidence intervals for each quantitative variables and population. Assuming a normal distribution of sample means, 商科统计代写 there is a 95% probability that the true mean of a population lies within the confidence interval shown for it.

Part 5

Hypothesis Testing

1) Variation of reading scores in group B and group C.

It appears from the boxplot that the variation among the reading scores in group B is less than that of in group C. Also, 商科统计代写 the standard deviation of reading scores in group B sample is 15.10, while the standard deviation of reading scores in group C sample is 18.14.

To see whether this is a statistically significant finding at the 5% level of significance, an F-stat was performed.

We use the “F-test two sample for Variances” command in Data Analysis. then we get the following table:

F-Test Two-Sample for Variances

	Variable 1	Variable 2
Mean	63.87	68.33
Variance	329.22	228.16
Observations	30	30
df	29	29
F	1.44
P(F<=f) one-tail	0.16
F Critical one-tail	1.86

The result show p-value for test is 0.16, so we don’t have enough evidence to reject the null hypothesis that reading scores of group B and group C’s variances are same at the 5% level of significance.

2) mean of reading scores in group B and group C.

Further, the mean value of reading scores in group B and group C respectively are 63.87 and 68.33. we want to see whether this is a statistically significant finding at 5% level of significance. Then a two-sample t-test is conducted.

By using the “t-Test: Two-Sample Assuming Equal Variances” command in Data Analysis, we can get the following result:

t-Test: Two-Sample Assuming Equal Variances

	Variable 1	Variable 2
Mean	63.87	68.33
Variance	329.22	228.16
Observations	30	30
Pooled Variance	278.69
Hypothesized Mean Difference	0
df	58
t Stat	-1.04
P(T<=t) one-tail	0.15
t Critical one-tail	1.67
P(T<=t) two-tail	0.30
t Critical two-tail	2.00

The results above show that the p-value one tail for testing is 0.15, also, we don’t have enough evidence to reject the null hypothesis that there is no significant difference between the mean reading score in group B and the mean reading score in group C.

3) Mean math score of group A and group C

Same as 1) and 2), 商科统计代写 we first conduct the variance testing using F-test, then using T-testing to test the mean of two sample.

The testing result are listed as below:

F-Test Two-Sample for Variances

	Variable 1	Variable 2
Mean	60.77	67.80
Variance	303.98	249.20
Observations	30	30
df	29	29
F	1.22
P(F<=f) one-tail	0.30
F Critical one-tail	1.86


t-Test: Two-Sample Assuming Equal Variances

	Variable 1	Variable 2
Mean	60.77	67.80
Variance	303.98	249.20
Observations	30	30
Pooled Variance	276.59
Hypothesized Mean Difference	0
df	58
t Stat	-1.64
P(T<=t) one-tail	0.05
t Critical one-tail	1.67
P(T<=t) two-tail	0.11
t Critical two-tail	2.00

The results shows that at the significance level 0.05, the mean math score of group B are equal with the mean score of group D.

Appendix-Sample Data

Group B

gender	race/ethnicity	parental level of education	lunch	test preparation course	math score	reading score	writing score
male	group B	some college	free/reduced	none	40	43	39
female	group B	some college	free/reduced	completed	65	75	70
male	group B	associate’s degree	standard	none	65	54	57
male	group B	some high school	standard	none	67	64	61
female	group B	associate’s degree	standard	none	47	49	50
female	group B	some college	standard	none	79	86	92
male	group B	high school	standard	completed	60	44	47
female	group B	high school	standard	completed	68	83	78
male	group B	some high school	standard	completed	61	56	56
male	group B	associate’s degree	standard	none	87	85	73
female	group B	some high school	standard	none	37	46	46
male	group B	associate’s degree	free/reduced	none	67	62	60
female	group B	some college	free/reduced	none	61	68	66
female	group B	bachelor’s degree	standard	none	61	72	70
female	group B	associate’s degree	standard	none	80	86	83
female	group B	high school	free/reduced	none	50	67	63
male	group B	associate’s degree	standard	completed	65	65	63
female	group B	associate’s degree	free/reduced	none	53	70	70
male	group B	some college	standard	completed	71	75	70
male	group B	high school	free/reduced	none	30	24	15
female	group B	bachelor’s degree	standard	completed	65	81	81
female	group B	some high school	free/reduced	completed	59	63	64
male	group B	high school	standard	completed	76	62	60
female	group B	high school	standard	completed	77	82	89
male	group B	associate’s degree	standard	completed	82	84	78
female	group B	high school	free/reduced	none	64	73	71
male	group B	high school	free/reduced	none	36	29	27
male	group B	high school	standard	none	82	82	80
male	group B	some college	free/reduced	completed	60	62	60
female	group B	high school	free/reduced	none	8	24	23

group C

gender	race/ethnicity	parental level of education	lunch	test preparation course	math score	reading score	writing score
female	group C	some college	free/reduced	none	76	83	88
male	group C	some high school	free/reduced	completed	53	37	40
female	group C	associate’s degree	standard	none	59	66	67
female	group C	some college	free/reduced	none	22	39	33
male	group C	some high school	free/reduced	completed	59	69	65
female	group C	some college	standard	completed	88	95	94
female	group C	some high school	standard	completed	85	92	93
male	group C	some high school	standard	none	51	52	44
male	group C	high school	standard	none	71	66	65
female	group C	high school	free/reduced	completed	67	79	84
female	group C	bachelor’s degree	standard	none	83	93	95
female	group C	some college	standard	completed	63	78	80
male	group C	some high school	free/reduced	none	69	71	65
male	group C	high school	standard	none	61	56	55
female	group C	some college	standard	none	84	87	91
male	group C	some college	free/reduced	none	68	68	61
female	group C	high school	free/reduced	none	36	53	43
male	group C	associate’s degree	standard	completed	51	60	58
male	group C	high school	standard	completed	75	69	68
female	group C	some college	free/reduced	completed	63	73	71
male	group C	some college	standard	none	53	39	37
female	group C	high school	standard	none	81	84	82
male	group C	high school	standard	none	62	67	58
female	group C	high school	standard	completed	60	64	74
male	group C	some high school	standard	none	64	58	51
male	group C	bachelor’s degree	standard	completed	63	64	66
male	group C	high school	free/reduced	none	54	72	59
female	group C	bachelor’s degree	free/reduced	none	67	75	72
male	group C	bachelor’s degree	free/reduced	completed	70	75	74
male	group C	high school	standard	none	81	66	64

Group D

mso-border-left-alt:none;border-right:1.0000pt solid windowtext;mso-border-right-alt:0.5000pt solid windowtext;

gender	race/ethnicity	parental level of education	lunch	test preparation course	math score	reading score	writing score
male	group D	associate’s degree	standard	none	40	52	43
male	group D	bachelor’s degree	free/reduced	completed	74	71	80
female	group D	some college	standard	none	69	74	74
male	group D	high school	standard	none	88	78	75
male	group D	some high school	free/reduced	none	45	37	37
male	group D	bachelor’s degree	free/reduced	completed	74	79	75
male	group D	some high school	free/reduced	completed	80	79	79
male	group D	some college	standard	none	76	64	66
male	group D	associate’s degree	standard	none	80	75	77
female	group D	associate’s degree	standard	completed	88	92	95
female	group D	bachelor’s degree	standard	none	59	70	73
female	group D	some college	standard	none	77	68	77
male	group D	associate’s degree	standard	none	71	66	60
female	group D	bachelor’s degree	standard	none	79	89	89
female	group D	associate’s degree	free/reduced	none	52	59	56
male	group D	some high school	free/reduced	none	62	49	52
female	group D	some college	standard	none	74	89	84
male	group D	high school	free/reduced	none	60	57	51
male	group D	high school	standard	none	46	34	36
male	group D	high school	standard	none	89	87	79
female	group D	high school	free/reduced	none	73	92	84
female	group D	some high school	free/reduced	completed	40	65	64
male	group D	bachelor’s degree	free/reduced	completed	39	42	38
male	group D	some high school	standard	completed	88	74	75
female	group D	bachelor’s degree	free/reduced	completed	93	100	100
male	group D	bachelor’s degree	free/reduced	none	50	42	48
male	group D	high school	free/reduced	none	74	70	69
female	group D	high school	free/reduced	completed	65	61	71
female	group D	some college	standard	none	62	70

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