## Mr. Felix K. Colon

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## MODEL ESSAY 3

STATISTICS ESSAY OUTLINE

DUE: MONDAY, DECEMBER 5, 2011

DOUBLE SPACED, 12 PT.

PARAGRAPH 1:  INTRODUCTION

1.     State the purpose “to show how statistics help people make decisions….”

2.     State that the essay will include problems covering measurements of central tendency, measurements of position, and measurements of dispersion.

3.     State examples of each type of measurements.

4.     State that the elements of thought will be used to critically think about problems involving statistics.

5.     Briefly describe problems that will be discussed.

PARAGRAPH 2: TREES

A forester measured 27 of the trees in a large wooded area where trees are up for sale.  He found a mean diameter of 10.4 inches and a standard deviation of 4.7 inches.  Suppose that these trees provide an accurate description of the whole forest and that a Normal model applies.

1.     Draw a graphical representation of the mean and 3+/- standard deviations for tree diameters.

2.     What would you expect the central 95% of all trees to be?

3.     About what percent of the trees should be less than an inch in diameter?

4.     About what percent of the trees should be between 5.7 and 10.4 inches in diameter?

5.     About what percent of the trees should be over 15 inches in diameter?

6.     If you were a buyer what would be your expectations (before you saw the trees) if you used the 68-95-99.7 Rule to determine approximately the number of trees out of 27 within 1, 2, and 3 standard deviations of the mean diameter? (show a graphical representation)

7.     Based on the point of view of a buyer’s estimates (who wants to resell the trees for Christmas), what would be the buyers conclusions and the consequences regarding a pricing strategy?

8.     What might be the assumptions of the buyer of the 27 trees? (Is the diameter of a tree the only important feature when buying a Christmas tree?)

9.     What would be the 84 percentile when referring to diameter of a tree?

PARAGRAPH 3: COMPARATIVE STUDY – 9th grade math regents scores from 5 schools

1.     Describe the data and list data for the five groups.

2.     Discuss measurements of central tendency for the 5 groups.

3.     Discuss measurements of position for the 5 groups.

4.     Discuss measurements of dispersion for the 5 groups.

5.     Discuss conclusions based on your data?

6.     Discuss possible assumptions made regarding the data.

7.     Discuss consequences based on your data?

8.     Discuss who might utilize this data (Use two Point of Views)?

9.     Draw box and whisker plot for each set of data. (separate sheet)

10. Draw graphic representation of the mean and data for 3 +/- standard deviations for each set of data. (separate sheet)

PARAGRAPH 4:  GATHERING DATA – relate the following to your survey

1.     Discuss collection process

2.     Discuss 2 types of data

3.     Discuss discrete/continuous data (quantitative data)

4.     Discuss univariate data

5.     Discuss sampling technique

6.     Discuss types of bias

PARAGRAPH 5:  GATHERING DATA  - SURVEY (the problem)

8.     Discuss measurements of central tendency for the class.

9.     Discuss measurements of position for the class.

10. Discuss measurements of dispersion for the class.

11. Discuss how the selection bias affects the reliability of the statistics in terms of a conclusion and consequences.

12. Assuming that the survey represents the population in general, what other assumptions can you make?

13. Assuming that the survey represents the population in general, from 2 points of view discuss possible conclusion and consequences.

PARAGRAPH 6: GATHERING DATA (additional questions)

20. What interval about the mean includes 95% of the data in the survey?

21. What interval about the mean includes 50% of the data in the survey?

SEPARATE SHEET:

22.  Draw a box and whisker plot to show the measurements of position.

23. Draw a graphical representation (number line) of the mean and 3 +/- standard deviations.

PARAGRAPH 7: CONCLUSION

1.     Re-state the purpose of the essay.

2.     Describe how the examples show how statistics help people make decisions? (Do this for each problem)

3.     Include in your description the elements of thought that you addressed.

4.     State the different types of fields or scenarios that can be addressed using  the 3 types of measurements (central tendency, position, dispersion).

5.     State that the data had to follow the features of normal distribution and state those features.

FACT SHEET:

GATHERING DATA

Data can be collected by conducting experiments where one group is the treatment group while the other is the control group not receiving any treatment.  In a double-blind experiment, neither the participant nor the researcher gathering the data know which group is the treatment group and which group is the control group.  You can also collect data through observational study. The value of the inferences that will be drawn from a set of data depends upon the quality of the data that has been collected.

Data Collection Process

1.     Determine the question to be studied.

2.     Decide what data to collect.

3.     Decide how to collect the data.

4.     Analyze the data.

Data can be of two general types qualitative or quantitative.  Qualitative data are descriptive (example: eye color, gender) represented by words.  Quantitative data are numerical.

Quantitative data can be classified as discrete (example: SAT scores, number of DVDs you own, number of TVs in your house) and Continuous (example: time spent on an activity, age).  Univariate data are values collected for a single variable (example: the age of the football fans at a game) and bivariate data are values collected for two different variables (example: the time spent studying and the GPA).

A population is the whole set of objects that are of interest (example: all the students of MVA) and a sample is a subset of the population (example: section A of the 9th grade).  It is important that the sample be representative of the entire population in order to gain reliable results.

Sampling Techniques:

Random sampling

1.     Selection made by a random process.

2.     Trustworthy results but costly.

3.     Data analysis is reliable but it can be difficult to identify every member of a population.

4.     Each member of the population has the same chance of being in the sample but supplementary information is important because of the random selection.

Convenience Sampling (EX: The Sleep Survey)

1.     Selected based upon expediency.

2.     Easy but limited useful applications.

3.     Accurate results when the members of the population are the same but sample not typically representative of the population and can lead to selection bias.

4.     Inexpensive but data unreliable.

Voluntary Response

1.     Selection determined by taking part in a survey.

2.     Easy but can be costly and time consuming.

3.     Survey can be completed at one’s convenience but can lead to non-response bias.

Types of Bias – Bias is an influence that affects the reliability of a statistical measurement in some way.

1.     Selection bias – a tendency within the sampling procedure that excludes one kind of person over another from the sample. (example: samples are not representative of the population)

2.     Non-response bias – respondents do not respond.

3.     Treatment bias – treatment and control group differ with respect to some factor other than the treatment where this other factor could bias the results.

4.     Measurement bias – where a measurement is consistently high or low.

MODEL ESSAY

STATISTICS

The purpose of this essay is to show how statistics help people make decisions in their personal or private lives.  This will include problems covering measurements of central tendency, measurements of position, and measurements of dispersion. I will discuss measurements of central tendency such as mean, mode, and median, measurements of position such as minimum, quartile 1-2-3, and maximum values, and finally, measurements of dispersion such as range and standard deviation.  I will use the elements of thought to critically think about the problems as they relate to statistics.  The problems that I will refer to are about trees, 9th grade math regents scores from 5 schools, and a survey of 11th graders and the hours slept on a particular Sunday night.

TREES

A forester measured 27 of the trees in a large wooded area where trees are up for sale.  He found a mean diameter of 10.4 inches and a standard deviation of 4.7 inches.  It is assumed that these trees provide an accurate description of the whole

MODEL ESSAY

STATISTICS

The purpose of this essay is to show how statistics help people make decisions in their personal or private lives.  This will include problems covering measurements of central tendency, measurements of position, and measurements of dispersion. I will discuss measurements of central tendency such as mean, mode, and median, measurements of position such as minimum, quartile 1-2-3, and maximum values, and finally, measurements of dispersion such as range and standard deviation.  I will use the elements of thought to critically think about the problems as they relate to statistics.  The problems that I will refer to are about trees, 9th grade math regents scores from 5 schools, and a survey of 11th graders and the hours slept on a particular Sunday night.

TREES

A forester measured 27 of the trees in a large wooded area where trees are up for sale.  He found a mean diameter of 10.4 inches and a standard deviation of 4.7 inches.  It is assumed that these trees provide an accurate description of the whole forest and that a normal model applies.

1.                      l          l           l           l           l           l           l

-3        -2         -1      MEAN    1          2          3  STANDARD DEVIATION

-3.7     1        5.7      10.4      15.1  19.8    24.5

2.  The central 95% of all the trees should be within 2 standard deviations or 1 inch and 19.8 inches.

3. 2.3% of the trees should be less than an inch in diameter. (0.1+0.5+1.7%)

4. 1 standard deviation or 34.1 % of the trees would be between 5.7 and 10.4 inches in diameter.

5. About 15.9% of the trees should be over 15 inches in diameter.

6.  .68(27) = 18.36 which means that 18 trees have diameters within 1 standard deviation, .95(27) = 25.65 which means that 26 trees have diameters with 2 standard deviation,  .997(27) = 26.919 which means that 27 trees have diameters within 3 standard deviations.

7.  The buyer of the 27 trees would conclude that the bigger the diameter the more he intends to charge.  With this strategy he could sell most of the trees for homes and the rest to establishments with large ceilings such as lobbies of large buildings or malls.

9.The assumptions of the buyer of 27 trees are that all the trees are freshly cut and healthy.

10. The 84th percentile ( at 1S.D. ) for diameters is 15.1 inches.

Comparative Study

1.     The data that was compared for analysis was 9th grade math regents scores from 5 schools.  For the measurement of central tendency: The low mean was 77 (school 1) and the high mean was 83.84 (school 3).  The low mode was 71 (school 2) and the high mode was 86 (school 1).  The low median was 78 (school 1, 5) and the high median 82 (school 3).

2.     For the measurements of position: The low minimum was 65 (school 1) and the high minimum 76 (school 3).  The low Q1 was 71 (school 2) and the high Q1 was 80 (school 3).  The low Q3 was 81 (school 5) and the high Q3 was 86 (school 3).  The low maximum was 87 (school 5) and the high maximum 95 (school 95)

3.     For the measurements of dispersion: The low range was 18 (school 1) and the high range was 29 (school 2).  The low standard deviation was 4.92 school 5) and the high standard deviation was 7.06 (school 2).

4.     Conclusion: School no. 3 is a high performing school because they received highest scores in mean, median, minimum, Q1, Q2, Q3, and maximum categories.  School no. 1 is a low performing school because they received lowest scores in the median, mean, minimum, and Q1.

In comparing the data from different schools, one assumes that the schools took exactly the same test, the students were given the same time to take the test, the tests were given on the same day, and the curriculum was the same so that they matched the questions on the tests.  As far as the consequences based on the data, the lower performing schools may have to consider implementing actions such as better preparation for the tests for the students, better training for the teachers, after-school programs to better prepare the students.   The better performing schools might continue with their teaching practices and possibly rewarding the teachers or the school for the great results.  From the point of view of a principal, changes might be necessary with the staff and how the students are taught if the school did poorly and maintaining the programs and rewarding the staff if the school did well.  From the point of view of a student or a teacher, the data might have reflected improvement from the past even if the school did not do well.

Central Tendency                    Position                                   Dispersion

 School Mean Mode Median Min. Q1 Q3 Max Range S.D. 1 77.4 86 78 65 73 78 88 23 7.03 2 78.3 71.8 80 66 71 84 95 29 7.07 3 83.84 80 82 76 80 86 95 19 5.47 4 77.74 75 79 66 75 84 86 20 5.67 5 77.78 76 78 66 75 81 87 21 4.92

SURVEY: “HOURS SLEPT THE NIGHT BEFORE”

A section of the 11th grade was surveyed and asked “how many hours did you sleep last night?” and the data collection process consisted of determining that question to be studied, deciding what data to collect, decide how to collect the data (performed in class) and analyzing the data.  There are two types of data, qualitative which is descriptive and quantitative which is numerical and the type of data that was collected.  The data was discrete, i.e. finite in number, collected in quarters of an hour such as 7 hours and 15 minutes or 6 hours and 45 minutes.  This is different from continuous data were the possibilities are infinite as in the fractions between 1 and 2. The data was univariate where data was collected for one variable (hours) as opposed to 2 variables that is called bivariate data. The sampling technique used was convenience sampling where a survey is taken at a location that is convenient and practical, such as this survey taken in the classroom.  Although this sampling technique is easy it has limited applications because the sample is not representative of the population and this can lead to a selection bias where every member of the population does not have an equal chance of being selected.

CLASS: HOURS SLEPT THE NIGHT BEFORE THE SURVEY

8          5          8          6          8          6          8          3          5          7          11        6

10        6          7          7          5          5          7          7          4          6          6.5       8

6          5

The data gathered was the hours slept the night before from students in the class.  The mean was 6.6 hours, the mode was 6 hours, and the median was 6.25 hours.  The minimum was 3, the maximum was 11, Q1 was 5, and Q3 was 8.  The range was 8 hours and the standard deviation was 1.7 hours.  It should be noted that since not every student in MVA was surveyed and that this was a convenience sampling, a selection bias makes the data unreliable if one is to use it to describe the hours slept the night before by all students in the school.  Therefore, one cannot make conclusions or discuss consequences about the number of hours slept for all MVA students. In order to make conclusions about the data, one has to assume that the hours surveyed are accurate and that these are hours are the number of hours that the students normally sleep.  From the point of view of parents and teachers the number of hours might be too short and a suggestion might be to sleep longer for healthier living and excellence in school.  From the point of view, the student might feel that the number of hours are adequate and have other factors affect their health and excellence in school.   The interval from 3.2 hours to 10 hours would include 95% of the data (95% of the data = +/- 2 standard deviations from the mean).  The central 50% would be from 5 hours to 8 hours (IQR = the middle 50% of the data = Q3=Q1).

BOX AND WHISKER PLOT:

_____________

O----  I          I              I------O

I_____I_______I

3       5       6.25          8        11

GRAPHICAL MODEL:

1.5       3.2       4.9       6.6       8.3       10        11.7

I           I           I           I           I           I           I           I           I

-3        -2         -1       MEAN    1          2           3           STANDARD DEVIATION

In conclusion, the purpose of statistics is to help people make decisions in their personal or professional lives by allowing the analysis of the data that could lead to conclusions and consequences.  The examples that were used were statistics relating to diameters of trees on average and their standard deviation for pricing strategy for example, 9th grade math regents scores from 5 schools to find the low and high performing schools, and the hours slept during the night by students on average to possibly be a factor in determining if it may affect health or productivity in the classroom.  In critically thinking about these issues, the 8 elements of thought were used.  The elements are purpose, concept, question, information, conclusion, assumption, consequences, and point of view.  I found that critically thinking about these problems, I realized that measurements of central tendency, position, and dispersion can be used in business, schools, and by government officials.  Finally, before conclusions regarding normal distribution can be made, the data had to have the features of normal distribution which are: (1) the data is unimodal, (2) relatively symmetrical, and follows the 65%-95%-99.7% rule.