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Statistical Analysis of a Given Data

See attached files.

Download the file entitled "elementary.xls." This file contains data on the 100 6th grade students at Oceanview Elementary School in Wichita, Kansas. Use this data to answer the questions below. Unless otherwise noted, use the 5% confidence level for all calculations.

1. Describing the Data
a) Identify each of the variables in the dataset and describe what type of variable they are (i.e. categorical, discrete numerical, ordinal, etc.). The easiest way to do this is make a list, top to bottom, of variables, their type, and your justification for calling them this type. DON'T FORGET TO justify your answers (i.e. why is this a "categorical" variable?).
b) For the numerical data, create a table of summary statistics. Include columns for AT LEAST the mean, standard deviation, median, minimum, and maximum. Define each of these terms (e.g. in statistics, what is a "mean?").

2. Sixth grade students are randomly put into classrooms every year. Three students, Michael Miller, Jose Rodriguez, and Scott Perez are best friends and want to be in the same class.
a) What is the probability that they are all in the same class?
b) What is the probability that they are all in Mr. Collins' class?
c) What is the probability that they are all in different classes?

3. There are 100 students overall, and they are distributed into 4 classrooms of equal size. There are 54 girls and 46 boys in sixth grade at Oceanview. Calculate the probability that:
a) A class will have 15 girls
b) A class will have 15 boys
c) A class will have more than 20 girls

4. Student Height and the Normal Distribution.
a) Construct a histogram of the students' height in centimeters.
b) Use the histogram and any other statistical tools that might be useful to determine if the distribution is normal. IMPORTANT! YOU MUST USE AT LEAST THREE PIECES OF EVIDENCE (any of the 6 we learned in class) TO DETERMINE WHETHER THE DISTRIBUTION IS NORMAL.
c) Construct a histogram of the students' height in inches.
d) Use the histogram and any other statistical tools that might be useful to determine if the distribution is normal. IMPORTANT! YOU MUST USE AT LEAST THREE PIECES OF EVIDENCE (any of the 6 we learned in class) TO DETERMINE WHETHER THE DISTRIBUTION IS NORMAL.
e) The data series are obviously related…the students' heights were measured in centimeters and that was converted to inches. What gives rise to any discrepancies you see, particularly in the shape of the histogram?

5. Testing for differences in student height.
a) The average height of 6th grade girls in the state of Kansas is 61.8 inches, and the average height of 6th grade boys in the state of Kansas is 59.4 inches. Are the heights of the boys and girls at Oceanview statistically different from the statewide averages? On what objective, statistical evidence do you base your answer here? IMPORTANT! PROVIDE THE EVIDENCE AND INTERPRET IT IN NARRATIVE.
b) Is there a statistically significant difference between male and female height among sixth graders at this school? On what objective, statistical evidence do you base your answer here? IMPORTANT! PROVIDE THE EVIDENCE AND INTERPRET IT IN NARRATIVE.

6. Teacher Quality
a) Calculate and compare the means of student test scores by class.
b) Test to see if the differences in test scores are statistically significant (remember to provide the evidence and INTERPRET the evidence, in narrative form. Then, if applicable, determine: Which teachers have better performing students than others? Which have worse performing students?
c) Can your evidence be used as conclusive evidence that some of the teachers are better/worse at teaching than others? Why or why not?
d) Repeat steps a and b using IQ scores instead of test scores. Does this change your response to part c? If so, how so? If not, why not?
e) The statewide average on the standardized test is 75. First, assume that you do not know the standard deviation for the population. How well does the school perform relative to the state average? How well does each teacher perform relative to the state average? Remember to provide objective, statistical evidence to back your claim, and INTERPRET the evidence, in narrative.
f) Repeat step e, this time using the fact that the statewide standard deviation of test scores is 8.
g) In parts e and f, correctly name the null and alternate hypotheses. Given those hypotheses, what would constitute a type I or a type II error? Did discovering the true population standard deviation in part f reveal any type I or type II errors in part e?

7. Intelligence and Test scores
a) Construct a scatter plot of IQ scores and test scores (use IQ as your independent variable and test score as your dependent variable).
b) Estimate the relationship between IQ and test scores using regression analysis
c) Interpret the meaning, both in the abstract and specifically related to this problem, of the Y intercept and beta coefficient. Are the results statistically significant? How do you know? On what evidence do you base your conclusion?
d) IQ is surely not the only determinant of student test scores. Estimate the relationship between class attendance, measured as days missing from class, and test scores.
e) Interpret the Y intercept and beta coefficient. Are the results statistically significant? How do you know? On what evidence do you base your conclusion?
f) Estimate test scores using BOTH IQ and attendance as independent variables. How do these relate to your answers to b-e?
g) Speculate: Are there other variables (i.e. ones not in the dataset) you think might be correlated with test scores? Why or why not? On what evidence do you base your claim here?

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Solution Summary

In this solution a number of questions related to probability and mathematical statistics are answered. These questions involves computation of summary statistics, calculation of probability, construction of histogram, testing normality using normal probability plot, testing the hypothesis regrading the means of populations, regression analysis and interpretation. Detailed answers to all the questions are provided with calculations in excel sheet.

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