Statistical Hypothesis Testing

mpx

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(a) State what is measured by the product moment correlation coefficient.

Hint

Solutions

(b) Describe in words the correlation in the following three diagrams, and suggest a value for

in each case:
(i)

(ii)

(iii)

2. A survey of weather in a tropical region is undertaken and the data analysed. The product moment correlation coefficient between the daily mean temperature and the daily rainfall is calculated as 0.267 with a

-value of 0.035 . Stating your hypotheses clearly, use a

significance level to test whether or not there is evidence of a positive correlation between the daily mean temperature and the daily rainfall in the region.
3. A sports scientist wishes to test whether those who are good at sprinting are also good at long jump. For a sample of 30 athletes, she records the time,

, taken for them to run 100 m , and the distance,

, jumped at long jump. She calculates the correlation coefficient,

, for

against

, and finds that

.
(a) Write down the null and alternative hypotheses for her test. Explain why the alternative hypothesis takes the form that it does.
(b) The critical value for a one-tailed test with

at

significance level is 0.3061 . Explain whether or not there is evidence for correlation in the population.

The manufacturer claims that packets of pasta have a mass of 500 g . The masses are normally distributed and the standard deviation is known to be 5 g . An inspector wants to find out if the packets are underweight. She weighs 16 packets of cereal and finds that they have a total mass of 7955 g . Carry out a hypothesis test at the level.
A driver wishes to improve his reaction time, which is a normally distributed random variable with mean 0.34 seconds and standard deviation 0.06 seconds. The driver undertakes a training course. Afterwards, a random sample of 8 reaction times is taken and the times are as follows (in seconds).

Assuming that the underlying standard deviation has not changed, test at the

level whether the driver’s mean reaction time has decreased after the training course.
6. The heights of three year old children are thought to be normally distributed, with a mean of 90 cm and a standard deviation of 5 cm . A scientist wishes to determine whether the mean height has changed, using a sample of 15 children.
(a) Stating your hypotheses clearly, find the critical region for a two-tailed test at the

significance level.
(b) A random sample of 15 children is taken and the following heights are measured:

88	94	90	92	91
89	93	87	95	90
96	93	89	91	93

Using your answer to (a), decide what conclusion the scientist should draw.

Another scientist repeats the experiment with a larger sample, and tests for a change in mean height at both the

level and the

level. She finds that at the

level she should reject the null hypothesis but at the

level she should not reject the null hypothesis. Explain, with reference to the two significance levels, how it is possible for the same data to give two different conclusions.

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