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Nuffield Free-Standing
Mathematics Activity
Gender
differences
© Rudolf Stricker
© Nuffield Foundation 2012
Manufacturers of children’s clothing need to
consider the body measurements of boys and
girls, and find answers to questions such as:
What are the mean values?
Are there significant differences between
the body measurements of boys and girls?
Do differences emerge at different ages?
Carrying out significance tests on mean values can help to
answer such questions.
In this activity you will use anthropometric data to carry
out significance tests of this type.
© Nuffield Foundation 2010
Distribution of the sample mean
When random variable X
follows a normal
distribution with mean μ
and standard deviation σ
the sample mean X
follows a normal
distribution with mean μ
and standard deviation
Think about
Why is the standard
deviation of the sample
mean smaller?
© Nuffield Foundation 2010
μ
μ – 3σ
μ + 3σ
x

n
 3 
n
μ
 3 
n
x
Summary of method for testing a mean
H0: mean, μ = value suggested
and the alternative hypothesis:
State the null hypothesis:
H1: μ ≠ value suggested (2-tail test)
or μ < value suggested or μ > value suggested (1-tail test)
x
Calculate the test statistic: z  
n
where x is the mean of a sample of size n
and σ is the standard deviation of the
population
© Nuffield Foundation 2010
Think about
Can you explain
this formula?
What do you do if
σ is not known?
Summary of method for testing a mean
Compare the test statistic with the critical value of z:
95%
For a 1-tail test
5% level, critical value = 1.65 or –1.65
5%
1% level, critical value = 2.33 or –2.33
–1.65 0
For a 2-tail test
95%
5% level, critical values =  1.96
1% level, critical values =  2.58
z
2.5%
2.5%
–1.96 0 1.96
z
If the test statistic is in the critical region (tail of the distribution)
reject the null hypothesis and accept the alternative.
© Nuffield Foundation 2010
Testing a mean: T-shirt example
A clothing manufacturer designs t-shirts for a chest circumference
of 540 mm. Is the mean for 4-year-old boys larger than this?
Null hypothesis
H0: μ = 540 mm
Alternative hypothesis H1: μ > 540 mm 1-tail test
Test statistic
z  x  
n 1
n
μ = 540
Think about
Why is a
1-tail test
used?
Using the data for 4-year-old males: x = 546.94355
z  546.94355 540 = 2.75
28.07432
124
© Nuffield Foundation 2010
σn – 1 = 28.07432 n = 124
Test statistic z = 2.75
For a 1-tail 1% significance test:
99%
1%
z
0
2.33 2.75
The test statistic, z, is in the critical region.
The result is significant at the 1% level,
so reject the null hypothesis.
Think about
Explain the
reasoning behind
this conclusion.
Conclusion
There is strong evidence that the mean is more than 540 mm.
© Nuffield Foundation 2010
Summary of method for testing the difference between means
State the null hypothesis:
H0: μA = μ B
(μ A – μB = 0)
and alternative hypothesis: H1: μ A ≠ μ B 2-tail test
or μ A < μ B or μ A > μ B
Calculate the test statistic:
Think about
Why are the variances added?
1-tail test
x x
A
B
z

 2 
 2
 A  B 


n
 n
A
B 


Compare with the critical value of z.
If the test statistic is in the critical region
© Nuffield Foundation 2010
reject the null hypothesis and accept the alternative.
Testing the difference between means: Hand length example
Using the data to test whether the hand lengths of 2-year-old
boys are significantly different from those of 2-year-old girls.
H0: μM = μ F
(μ M – μF = 0)
H1: μM ≠ μ F
2-tail test
x x
M F
Test statistic: z 

2 
2 

 (n - 1)M

 (n - 1)F 



n
n


M
F


© Nuffield Foundation 2010
Testing the difference between proportions: Example
Using the hand length data for 2-year-olds on the spreadsheet gives:
x = 103.25 mm
M

x = 101 mm
F

= 7.9332 mm
nM = 52
(n 1)F = 5.1478 mm
nF = 49
(n 1)M
x x
M F
Test statistic: z 

2 
2 

 (n - 1)M

 (n - 1)F 



n
n


M
F


z
103.25101
= 1.70


2
2
 7.9332
5.1478 




52
49




© Nuffield Foundation 2010
Testing the difference between means: Hand length example
Test statistic: z = 1.70
For a 2-tail 5% test
95%
2.5%
2.5%
1.7
– 1.96 0 1.96
z
The test statistic is not in the critical region.
Conclusion
There is no significant difference
between the hand lengths of
2-year-old boys and girls.
© Nuffield Foundation 2010
Think about
Explain the reasoning
behind this conclusion.
At the end of the activity
• What are the mean and standard deviation of the distribution
of a sample mean?
• Describe the steps in a significance test for a mean value.
• Describe the steps in a significance test for the difference
between means.
• When should you use a one-tail test and when a two-tail test?
• Would you be more confident in a significant result from a 5%
significance test or a 1% significance test? Explain why.
© Nuffield Foundation 2012