F.M. Statistics Analysis Task 3.1 PracticeName: ______

Practice SAC

The study of height is known as auxology. Growth and height have long been recognised as the measure of the health and wellness of individuals. Adult heights between ethnic groups often differ significantly and the average height for each sex within a country's population is significantly different.

The average height (mean) of males and females in 100 countries are shown on Data Sheet 1. The heights are in cm.

Part A (28 marks)

In this section you will analyse male and female heights from different countries.

A1. Randomly select 20 countries from Data Sheet 1 and record the male and female heights in the table below.(THIS HAS ALREADY BEEN DONE FOR YOU TO SAVE TIME!)

No. / Country / Male Height (cm) / Female Height (cm) / No. / Country / Male
Height (cm) / Female Height (cm)
6 / 179.2 / 166.0 / 99 / 167.0 / 156.0
73 / 174.0 / 160.0 / 72 / 157.5 / 162.0
2 / 169.8 / 161.5 / 32 / 157.5 / 142.2
43 / 174.2 / 160.8 / 25 / 172.0 / 166.8
31 / 177.9 / 171.0 / 95 / 173.7 / 160.4
98 / 170.5 / 164.8 / 5 / 178.4 / 163.9
3 / 175.0 / 166.0 / 12 / 162.5 / 155.7
84 / 180.0 / 166.9 / 70 / 178.3 / 168.5
62 / 163.8 / 157.8 / 64 / 176.4 / 164.8
21 / 180.6 / 169.3 / 20 / 180.3 / 167.3

[2 marks]

A2a. Fill in the following table for the heights of males and females for your selected countries.

Height (cm) / Number of Males / Number of Females / Percentage of Males / Percentage of Females
/ 0 / 1 / 0% / 5%
/ 1 / 3 / 5% / 15%
/ 5 / 15 / 25% / 75%
/ 11 / 1 / 55% / 5%
/ 3 / 0 / 15% / 0%
Totals: / 20 / 20 / 100 / 100

b. Draw a segmented bar graph on the grid below to illustrate the data in the table using the given key.

c. Comment on the difference between male and female heights.

Majority of the students are in the range 160-180: 75% of the females are 160-170 cm tall compared to 55% of the males who are 170-180 cm tall. Males are taller in general.

15% of the males are above 180 cm compared to 0% females.

5% of the females are 140-150 cm tall compared to 0% males.

[4 + 4 + 1 = 9 marks]

A3. Determine the 5 Number Summary Statistics for the male and female heights, giving your answers to 4 significant figures.

5 Number Summary Statistics / Male Height / Female Height
Min Height / 157.5 / 142.2
Lower Quartile / 168.4 / 160.2
Median / 174.1 / 164.4
Upper Quartile / 178.4 / 166.9
Max Height / 180.6 / 171.0

[2 marks]

A4.a. Calculate the outlier limits for Male and Female Heights, giving your answers to 4 significant figures. Identify and explain any outliers.

Male Heights
Male Hts
IQR / 10.0
1.5*IQR / 15.0
Q1-1.5*IQR / 153.4
Q3+1.5*IQR / 193.4
Outliers? / NO
/ Female Heights
Female Hts
IQR / 6.7
1.5*IQR / 10.1
Q1-1.5*IQR / 150.2
Q3+1.5*IQR / 177.0
Outliers? / YES
Height 142.2 cm is an outlier.

[4 marks]

A4 b. Construct parallel box plots for the Male Heights and Female Heights data.

A5. Calculate the mean and standard deviation of the data, correct to 1 decimal place.

Male / Female
Mean / 172.4 / 162.6
Standard Deviation / 7.3 / 6.4

[2 marks]

A6 a. Assuming the male heights form an approximate normal distribution, calculate the male height that is 1 standard deviation below the mean height.

1 s.d. < mean: 165.1

b. What percentage of males have a height above the value you have found in part A6a?

% age > :84%

c. Assuming the female heights form an approximate normal distribution, calculate the female height that is 2 standard deviations above the mean height.

2 s.d > mean: 175.4

d. What percentage of females have a height below the value you found in part A6c?

% age > 97.5%

e. Fred is a male adult whose height is 165cm. Convert his height to a z score, giving your answer to 1 decimal place, and comment on how extreme his height is.

z-score: -1.0

[1 + 1 + 1 + 1 + 2 = 6 marks]

A7. Compare your sample data for the countries you have chosen. In particular, discuss the centre and spread of the heights of males and females. Your comments should include the relevant statistics you have calculated.

Centre:

The mean height of the females is 162.6cm compared to males 172.4cm. The median height of females is 164.4 cm compared to males 174.1cm. The males are generally taller than females.

Spread:

The IQR of the heights of the females is 6.7cm compared to males 10.0 cm. The standard deviation of the heights of females is 6.4cm compared to males 7.3 cm. The males are generally taller than females. The male height is more spread compared to female height.

Outliers:

The male data do not have any outliers but the female data does have an outlier of 142.2 cm. a very short female.

[3 marks]

Part B (27marks)

In this section you will determine if the male height is a good predictor of the female height.

B1aRandomly select another 10 countries from Data Sheet 1 and record the male and female heights in the table below.(THIS HAS ALREADY BEEN DONE FOR YOU TO SAVE TIME!)

No. / Country / Male Height (cm) / Female Height (cm)
40 / 177.0 / 165.8
12 / 162.5 / 155.7
25 / 172.0 / 166.8
86 / 173.0 / 161.7
38 / 174.2 / 160.0
60 / 184.8 / 168.7
8 / 165.1 / 154.7
70 / 178.3 / 168.5
99 / 167.0 / 156.0
55 / 169.0 / 159.0

b. Construct a scatterplot of the female height data against the male height data that you selected in question B1a. Take the male height as the explanatory variableand the female height as the response variable.

Use an appropriate scale on both axes and label the axes clearly.

[2 +4 = 6 marks]

B2a. Determine the value of Pearson's correlation coefficient, to 4 significant figures for the data chosen in B1a.

=0.8870______

b. Using the scatter plot in B1b and this r value, comment on the relationship between the male and female heights in terms of form, strength and direction.

Strong, Positive and Linear

c. Find the coefficient of determination,,to 4 significant figures.

=0.7868______

d. Interpret in terms of the female and male heights.

78.6% of the variation in Female Height can be explained by the variation in the Male Height. 21.4% is due to other factors.

[1 + 3 + 1 + 2 = 7]

B3 a. Determine the least squares regression line for the data. Give your coefficients to 4 significant figures.

Female height = 37.960.7181Male height

b. Using the least squares regression equation, predict the female height when the male height is 165cm. Give the answer to 4 significant figures.

156.4 cm

c. Using the least squares regression equation, predict the female height when the male height is 185 cm. Give the answer to 4 significant figures.

170.8 cm

d. Using the data above accurately plot the two points in the scatter plot, labelling the points

A and B. Draw a line through the two points.Refer to the Scatter Graph on page 6.

f. Interpret the slope (gradient) of the Least Squares Regression line in terms of the variables.

As male height increases by 1 cm the female height increases by 0.7181 cm.

g. Using the Least Squares Regression equation, predict the female height when the male height is 165cm. Give the answer to 4 significant figures.

156.4 cm

h. Comment on the reliability of your prediction in B3g.

It is reliable since there is data to support the prediction. It’s an interpolation.

[1 + 3 + 2 + 2 + 2 + 2 + 1 + 1= 14 marks]

Section C (15 marks)

A study was conducted to record the relationship between the growth of human cells over a time interval. The data is shown in the table below:

Time in Minutes / 10 / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / 100
Number of Cells / 30 / 50 / 80 / 130 / 210 / 340 / 550 / 890 / 1440 / 2330

A scatter plot of the data above is shown below:

C1a From the scatter plot it appears that a linear model is not the best model.To confirm your suspicions, you decide to plot the residuals. Based on the least squares regressionline, generate a table of residuals for the scatterplot shown, giving the residuals to the nearest whole number.

Time in minutes / Residuals
10 / 403.3
20 / 205.9
30 / 18.5
40 / -148.9
50 / -286.3
60 / -373.7
70 / -381.1
80 / -258.5
90 / 74.2
100 / 746.8

b. Construct a residual plot for the data in C1a, using an appropriate scale on both axes. Label the axes clearly.

c. Comment on the pattern of residuals and suggest whether a transformation of data would be appropriate

The residual plot has a parabolic pattern suggesting that the data is non-linear therefore a transformation of data is appropriate to linearize the data.

d. You decide to apply a transformation to the value (the number of cells). Complete the table below using this transformation, giving the values to 2 decimal places.

Time in Minutes / 10 / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / 100
Number of Cells / 30 / 50 / 80 / 130 / 210 / 340 / 550 / 890 / 1440 / 2330
/ 1.48 / 1.70 / 1.90 / 2.11 / 2.32 / 2.53 / 2.74 / 2.95 / 3.16 / 3.37

e. Using your calculator, create a scatter plot of data against the time. Why is this transformation a better model?

f. Find the least squares regression equation of the non-linear model, giving the coefficients to 4 significant figures.

__1.275____0.02094__ time

g. Use your equation to predict the number of cells after 120 minutes. Give your answer to the nearest whole number.

[2 + 4 + 2 + 2 + 1 + 2 + 2 = 15 marks]

ACCURACY (Correct rounding of numbers and using specified number of decimal places) [2 marks]

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