Download Domain: Fitting a normal distribution to given data

Student Worksheet: “Are You Fit?”
Class: _____
Level:
Name: _______________ (
)
Form 6 or 7 (AS level)
Domain: Fitting a normal distribution to given data
Objectives:
To fit a normal distribution to given data by using
(i) frequency polygon,
(ii) normal plot
Prerequisite Knowledge:
1. Basic knowledge of statistical graphs
2.
Transform a normal distribution to N(0, 1) and the use of standard normal table.
Preparation:
Gather height and weight data (form, gender, age, weight, height) for S.6 and
S.7 students in your school using an Excel spreadsheet, or
you can use the data file ‘Height&Weight115’ provided.
Activity:
1. Think of the criteria of measuring body fitness.
2. Do you consider yourself ‘fit’? Why or why not?
Useful Web sites:http://www.ryvita.co.uk/
http://www.room42.com/nutrition/basal.shtml
http://n.ethz.ch/student/cschlup/convert.html
(on-line height/weight calculator)
(basal metabolism calculator)
(body height and weight converter)
Main Task:
1. Record the weight (in kg) of all students in the frequency table. [ Note: 1 lb = 0.454 kg ]
Weight (kg)
Frequency
Below 40
40 – 44
45 – 49
50 – 54
55 – 59
60 – 64
65 – 69
70 – 74
75 – 79
80 – 84
85 or above
Total
(a) Compute the mean [ =
kg] and standard deviation [ =
kg] of the distribution
(b) Using the model N(, 2), find the probability (using the normal table provided) and
expected frequency (= probability × total frequency) in the following table:
Class Interval
Observed frequency
Probability
Expected frequency
34.5 – 39.5
39.5 – 44.5
44.5 – 49.5
49.5 – 54.5
54.5 – 59.5
59.5 – 64.5
64.5 – 69.5
69.5 – 74.5
74.5 – 79.5
79.5 – 84.5
84.5 – 89.5
Total
(c) Construct two frequency polygons, one for the original distribution (using observed
frequencies) and one for the fitted normal distribution (using expected frequencies), on
the same graph.
(d) Does the result show that the students’ weight distribution is normal?
(e) Consider the table below and complete the columns: cumulative frequency, relative c.f.
(cumulative frequency), and the value of z (using the normal table provided).
Upper boundary, xU Cumulative frequency Relative c.f.   (z)
z
39.5
44.5
49.5
54.5
59.5
64.5
69.5
74.5
79.5
84.5
1.0000
89.5

(f)
Construct the scatter plot of z against xU (except for the last value).
(g) Is the graph close to a straight line (i.e., normal distribution is a good fit)?
If ‘yes’, estimate the values of  and  from the graph
[Hint: x-intercept gives an estimate of  and the slope gives 1/]
2.
Record the height (in cm) of all students in the frequency table. [ Note: 1 inch = 2.54 cm ]
Height (cm)
Frequency
Below 150
150 – 154
155 – 159
160 – 164
165 – 169
170 – 174
175 – 179
180 – 184
185 or above
Total
(a) Compute the mean [ =
cm] and standard deviation [ =
cm] of the distribution
(b) Using the model N(, 2), find the normal probability and expected frequency:
Class Interval
Observed frequency
Probability
Expected frequency
144.5 – 149.5
149.5 – 154.5
154.5 – 159.5
159.5 – 164.5
164.5 – 169.5
169.5 – 174.5
174.5 – 179.5
179.5 – 184.5
184.5 – 189.5
Total
(c) Construct the two frequency polygons (observed / expected) on the same graph.
(d) Does the result show that the students’ height distribution is normal?
(e) Complete the columns: cumulative frequency, relative c.f., and the value of z .
Upper boundary, xU Cumulative frequency Relative c.f.   (z)
z
149.5
154.5
159.5
164.5
169.5
174.5
179.5
184.5
1.0000
189.5

(f)
Construct the scatter plot of z against xU (except for the last value).
(g) Is the graph close to a straight line?
(If ‘yes’, estimated the values of  and  .)
3.
Compute the Body-Mass Index (BMI) for each student:
BMI = [weight (kg)] / [height (m)]2
Some specialists in nutrition would use this index as a measure of body fitness.
Related Web sites for Body-Mass Index:http://www.cc.ysu.edu/~doug/hwp.cgi
http://www.kcnet.com/~marc/bmi.html
http://www.fitnesstutor.com/bmi2.html
http://augusta.webpoint.com/health/bmicalc.htm
(plus waist-to-hip ratio)
Or, you can search for about 30,000 Web pages for body mass ratio:http://ink.yahoo.com/bin/query?p=body+mass+ratio&hc=0&hs=0
Record the BMI of all students in the frequency table.
BMI
Frequency
Below 18.0
18.0 – 18.9
19.0 – 19.9
20.0 – 20.9
21.0 – 21.9
22.0 – 22.9
23.0 – 23.9
24.0 – 24.9
25.0 – 25.9
26.0 – 26.9
27.0 or above
Total
(a) Compute the mean [ =
] and standard deviation [ =
] of the distribution
(b) Using the model N(, 2), find the normal probability and expected frequency:
Class Interval
Observed frequency
Probability
Expected frequency
16.95 – 17.95
17.95 – 18.95
18.95 – 19.95
19.95 – 20.95
20.95 – 21.95
21.95 – 22.95
22.95 – 23.95
23.95 – 24.95
24.95 – 25.95
25.95 – 26.95
26.95 – 27.95
Total
(c) Construct the two frequency polygons (observed / expected) on the same graph.
(d) Does the result show that the students’ BMI distribution is normal?
(e) Complete the columns: cumulative frequency, relative c.f., and the value of z .
Upper boundary, xU Cumulative frequency Relative c.f.   (z)
z
17.95
18.95
19.95
20.95
21.95
22.95
23.95
24.95
25.95
26.95
1.0000
27.95

(f)
Construct the scatter plot of z against xU (except for the last value).
(g) Is the graph close to a straight line?
(If ‘yes’, estimated the values of  and  .)
Exercise:
1.
Repeat tasks 1 to 3 above for :
(a) male students only,
(b) female students only.
2.
Compare the results obtained and summarize your findings.
Enrichment:
Try to use available data such as monthly income, pocket money, shortsightedness, etc (or you can
collect some continuous data yourself) and fit an appropriate normal distribution to the data.