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Transcript of © Nuffield Foundation 2012 Nuffield Free-Standing Mathematics Activity Successful HE applicants ©...
© Nuffield Foundation 2012
Nuffield Free-Standing Mathematics Activity
Successful HE applicants
© Rudolf Stricker
© Nuffield Foundation 2010
Are 50% of the applications for courses in higher
education from females?
Are the same proportion of male and female applicants
successful?
Carrying out significance tests on proportions, and the difference between proportions, can help to answer such questions.
In this activity you will use data on successful applications for courses in higher education to carry out significance tests of this type.
© Nuffield Foundation 2010
Distribution of a sample proportion
p
ps
n
pqp 3
If the sample size, n, is large (>30)the sample proportion, ps, will follow a normal distribution
n
pqand standard deviation
with mean p
n
pqp 3
where q = 1 – p
Think aboutWhy is it important that the sample is large?
© Nuffield Foundation 2010
Calculate the test statistic:
npq
pspz
Summary of method for testing a proportionState the null hypothesis:
H0: population proportion, p = value suggested
and the alternative hypothesis:
H1: p ≠ value suggested (2-tail test)
or p < value suggested or p > value suggested (1-tail test)
Think aboutCan you explain this formula?
where ps is the proportion in a sample of size n and q = 1 – p
© Nuffield Foundation 2010
If the test statistic is in the critical region (tail of the distribution)
Compare the test statistic with the critical value of z:
Summary of method for testing a proportion
For a 1-tail test
1% level, critical value = 2.33 or –2.33
reject the null hypothesis and accept the alternative.
5% level, critical value = 1.65 or –1.65
For a 2-tail test
5% level, critical values = 1.96
1% level, critical values = 2.58
-1.65 z0
5%
95%
95%
z0 1.96
2.5%
-1.96
2.5%
© Nuffield Foundation 2010
Testing a proportion: Example
In 2010 a newspaper article said that the proportion of people accepted on higher education courses over 20 years old was 16%.
Null hypothesis H0: p = 0.16
Alternative hypothesis H1: p < 0.16
Test statistic
= –2.77
npq
pspz
From 2010 data:
q = 1 – p
p = 0.16
= 0.84
415 150.840.160.160.1518
z
n = 15 415
1-tail test
Using 2010 data to test this percentage:
Think aboutWhy is a 1-tail test used?
= 0.1518 415 152340 ps
© Nuffield Foundation 2010
Test statistic = –2.77
–2.77
The test statistic, z, is in the critical region.
The result is significant at the 1% level, so reject the null hypothesis.
For a 1-tail 1% significance test:
ConclusionThere is strong evidence that the proportion reported is too high.
npq
pspz
–2.33 z0
1%
99%
Think aboutExplain the reasoning behind this conclusion.
© Nuffield Foundation 2010
samples in number Totalattribute with itemsof number Totalp
Calculate the test statistic:
Compare with the critical value of z.
State the null hypothesis: H0: pA = pB
H1: pA ≠ pB
Summary of method for testing the difference between proportions
and alternative hypothesis:
or pA < pB or pA > pB
2-tail test
1-tail test
where
BA
SBSA
nnpq
ppz
11 q = 1 – p
(pA – pB = 0)
Think aboutExplain the formula for the test statistic. If the test statistic is in the critical region
reject the null hypothesis and accept the alternative.
pSA , pSB , nA and nB are values from the samples.
© Nuffield Foundation 2010
Using 2010 data to test whether the proportion of males that were accepted is equal to the proportion of females that were accepted.
Test statistic:
H0: pM = pF (pM – pF = 0)
H1: pM ≠ pF 2-tail test
Testing the difference between proportions: Example
FM
SFSM
nnpq
ppz
11
© Nuffield Foundation 2010
Testing the difference between proportions: Example
In the 2010 sample: 7062 out of 45 455 males and 8353 out of 54 030 females were successful.
030 54 455 458353 7062
p = 0.154 948
q = 1 – 0.154 948 = 0.845 052
455 457062SMp = 0.155 362
030 548353SFp = 0.154 599
Test statistic:
FM
SFSM
nnpq
ppz
11
030 541
455 451052 0.845948 0.154
599 0.154362 0.155z = 0.331
© Nuffield Foundation 2010
z = 0.331
95%
z0 1.96
2.5%
- 1.96
2.5%
0.331
Conclusion There is no significant difference between the proportion of males and the proportion of females accepted.
The test statistic is not in the critical region.
Think aboutExplain the reasoning behind this conclusion.
For a 5% testTest statistic:
Testing the difference between proportions: Example
At the end of the activity
• What are the mean and standard deviation of the distribution of a sample proportion?
• Describe the steps in a significance test for a sample proportion.
• Describe the steps in a significance test for the difference between sample proportions.
• 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