1. LBSRE1021 Data Interpretation Lecture 9
Hypothesis Testing
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

2. Last week- Confidence Interval
With a sample size n>30 we can estimate the range within which the population mean lies (the confidence interval for the mean) by:
pop mean = x +/ se (95% CI)
pop mean = x +/ se (99% CI)
se = sd for sample/ /n
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

3. Testing a mean What happens if we know the value of some
population mean and a sample taken from this population doesn’t appear to agree with it?
Either
the population parameter is not what we think, or
the difference between the sample and population is simply due to chance effects
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

4. The null hypothesis
We make the hypothesis that any departure from the supposed true population parameter by the sample estimate is due to chance effects
We then see if we can disprove this hypothesis at a particular significance level
The null hypothesis is written as H0
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

5. The alternative hypothesis
If we reject the null hypothesis we automatically accept the alternative hypothesis, H1
This hypothesis can be two-tailed or one-tailed
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

6. Types of tests The form of the test depends on: the size of the sample
the number of samples
whether it is the mean or proportion that is of interest
For tests on the mean we use the Z-statistic
for large samples and the t-statistic for small
samples
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

7. Large (single) sample test for a mean
When the sample size is large we use the normal approximation
The test can be one-sided or two-sided
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

8. A two-tailed test (not equal to) using a 5% significance level
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

9. One-tailed test (less than) using a 5% significance level
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

10. One-tailed test (greater than) using a 5% significance level
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

11. Steps for carrying out a hypothesis test
Set up the null and alternative hypotheses
Decide on the significance level (normally 5%) and determine the critical values
Calculate the test statistic (this is the number of standard deviations that the sample estimate is away from the population parameter)
Decide whether to accept or reject the null hypothesis
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

12. Example 1
A food manufacturer processes and cans baked beans. The net weight of a standard can of beans is supposed to be 420g but a random sample of 50 cans gave an average weight of 415g with a standard deviation of 30g. Is there any evidence that the true mean weight is not 420g?
(Use a 5% significance level)
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

13. Solution to Example 1
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

14. Conclusion of the test result
As the test statistic is between –1.96 and 1.96
we cannot reject H0. That is, there is no
evidence at the 5% level of significance that
the mean is not 420g.
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

15. Example 2 (1 of 3) Bags of sugar weighing 1000g
Weight packed by machine drifts
Need to detect drift and reset machine but not stop process unnecessarily
Null hypothesis u = 1000
ie:
H0 : u = 1000
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

16. Example 2 (2 of 3) Standard deviation = 50g For 100 bags in a sample:
s.e. = s.d. = 50 = 5g
n 10
Sample mean x to test H0
Z = x - u = x
s.e
Test is two-sided so compare value of Z with 1.96 and 2.576
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

17. Example 2 (3 of 3) e.g. if sample mean 1015g Z = 1015-1000 = 3 5
This is > (1% level of significance)
H0 rejected, H1 : u = 1000 accepted
Mean weight now over 1kg with < 1% chance of being wrong to conclude this
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

18. Example 3 (1 of 3) Noise in street
Measured over long time between 4:30pm and 5:30pm
Average noise decibels
s.d decibels
Residents take 50 readings
Mean 134 decibels
Is noise increasing?
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

19. Example 3 (2 of 3) H0 : u = 130 H1 : u > 130
(Is test one or two-sided?)
Z = | x - u | =
s.e / 50
= 1.41
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

20. Example 3 (3 of 3) Critical values of Z 1.645 (5%) and 2.326 (1%)
Difference between population mean and sample mean not significant as
Z = 1.41 < 1.645
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

21. t-test for a sample mean
When the sample size is small we can use the t-test.
The procedure is the same except that the critical value is found from t-tables using n–1 degrees of freedom.
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

22. Hypothesis tests involving two means
We can have:
two large independent samples
two small independent samples
paired samples
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

23. Two large independent samples
The standard error of the difference between
the means is:
The test statistic is:
Z =
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

24. Two large independent samples continued
The null and alternative hypotheses are:
H1: 1 - 2  0 for a two sided test and
H1: 1 > 2 or 1 < 2 for a one sided test
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

25. t-test for two independent small samples
The standard error for two small samples is: =
where is the estimate of the pooled standard deviation of the populations and is given by:
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

26. t-test for two independent small samples continued
The test statistic is given by:
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd

27. Paired samples The null hypothesis is that the difference of the
population means is zero and the alternative
hypothesis can be either one-tailed or two-tailed.
The test statistic for this test is:
Where
Essential Quantitative Methods 2nd edn © Les Oakshott Palgrave Publishers Ltd