Revision

1
Revision

• Ideas of hypothesis testing
• One-sample t-test
• Two independent samples t-test
• Paired t-test

2
Stages in statistical analysis
3
Example 1

• Established records show that the lengths of
  mussels from an estuary are normally distributed
  with a mean of 30 mm. A random sample of 25
  mussels was taken from a polluted beach.
• Is there sufficient evidence to conclude that
  pollution appears to inhibit the growth of
  mussels?

4
Data

• 26.72 16.44 18.72 41.39 21.31
• 29.87 19.06 17.03 15.29 23.95
• 18.15 31.22 26.33 34.88 17.57
• 21.02 26.45 25.56 21.32 30.40
• 19.12 42.56 39.63 35.12 25.95

5
Display the data
6
One-sample t-test

• Null hypothesis pollution has no effect.
• Mean length has not changed.
• H0 ? 30
• Alternative hypothesis pollution inhibits growth
• Mean length is less than previously
• H1 ? lt 30
• P-value What is the probability of obtaining a
  mean of 25.76 or less if the true mean is 30?

7
Analysegt Compare means gt One-sample t
8
Output
9
Conclusion

• Always state the final conclusion in terms of the
  original problem and avoiding statistical jargon.
• There is evidence to suggest that pollution does
  inhibit the growth of the mussels.
• The mussels from the polluted area are 4.24cm
  smaller on average.
• 95 confidence interval for mean length of
  polluted mussels is (22.42, 29.11)

10
Assumptions

• Test assumes that the data are from a normal
  distribution.
• Check using a Q-Q or normal plot.

11
Example 2

• Data were collected to test the hypothesis that
  the size of cuckoos eggs differ depending on the
  type of nest in which they lay their egg.
• Samples of cuckoos eggs from the nests of hedge
  sparrows and reed warblers were obtained.

12
Choice of test

• The two sets of data are independent of each
  other so the appropriate test to examine the
  difference in the means of the two types of eggs
  would be a independent or two-sample t-test
• Assuming both sets of data come from normal
  distributions.

13
Boxplots
14
Hypotheses

• H0 ?1 - ?2 0
• H1 ?1 - ?2 ? 0
• using a two-tailed alternative

15
Analysegt Compare means gt Independent- samples t
test
16
Summary statistics
17
Results of hypothesis test
Note equal variances should not be assumed
18
Conclusion

• Always state the final conclusion in terms of the
  original problem and avoiding statistical jargon.
• There is evidence that the size of cuckoo egg
  differs depending of the type of nest. The eggs
  in hedge sparrow nests are larger that those in
  reed warbler nests.
• How much larger?

19
Confidence interval for the difference in two
means
Suggests eggs in hedge sparrows nests are between
0.45 and 1.53 mm longer, on average, than those
in reed warblers nests.
20
Assumptions

• Test assumes that the data are from a normal
  distribution.
• Check using a Q-Q or normal plot for each group.
• Use test assuming equal variance, if possible.
• Not possible for this example.

21
Example 3

• Does smoking increase the aggregation of
  platelets in the blood?
• Aggregation measured in 14 individuals before and
  after smoking a cigarette

22
Data
23
Choice of test

• Two results obtained from each individual.
• Data for before and after are clearly not
  independent.
• Paired t-test is appropriate.

24
Hypotheses

• H0 ?d 0
• H1 ?d gt 0
• using a one-tailed test

25
Display the data
26
Analysegt Compare means gt Paired samples t test
27
Results
28
Null hypothesis

• Often the aim of an experiment is to look for a
  difference or an improvement.
• Null hypothesis is always that there is no
  difference.
• A significant result means that this hypothesis
  of no difference is rejected.
• If a result is not significant then the null
  hypothesis is not rejected.

29
Hypothesis test

• Do the sample means differ from the hypothesised
  mean by
• A small amount that can be explained by sampling
  error
• A large amount indicating the null hypothesis of
  no difference is wrong

30
P-value (sig)

• Degree of difference measured by the p-value, the
  probability of obtaining the sample mean if the
  hypothesised mean is correct.
• Small difference, large p-value
• Large difference, small p-value

31
Confidence interval

• Given inherent error due to the sampling process,
  what range of values seem feasible for the mean
  of the population?
• Can be used for
• Estimating the mean with no hypothesis test
• Checking a hypothesised value
• Estimating the mean when the null hypothesis has
  been rejected