STAT170 Workshop Notes prepared by Nan Carter for Numeracy ...
STAT170 Workshop Notes prepared by Nan Carter for Numeracy ...
STAT170 Workshop Notes prepared by Nan Carter for Numeracy ...
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EXERCISE 3. The heights of the adult males in a particular<br />
country are known to be normally distributed with a mean of<br />
1.7 metres and a standard deviation of 0.09 metres. Find the<br />
probability that<br />
• an individual man, randomly selected, has a height of less<br />
than 1.5 metres.<br />
• a random sample of 25 men has average height of more than<br />
1.72 metres.<br />
**** In all the above exercises the distribution of the variable of interest <strong>for</strong><br />
individuals in the parent population was ‘normal’. What happens when it is not<br />
normally distributed (or when its distribution is unknown)? There is a theorem that<br />
proves that if the samples are large enough then the sampling distribution of means is<br />
approximately normal (Central Limit Theorem). At Macquarie we can apply the CLT if<br />
n>25. This is a most important and useful result: we can always get a normal variable <strong>by</strong><br />
taking a large-enough sample and working with the sample mean.<br />
EXERCISE 4. A certain brand of rope is known to have a<br />
breaking strength on average of 25 kg with a standard<br />
deviation of 0.5 kg. A random sample of 50 pieces of rope is<br />
tested <strong>for</strong> breaking strength. What is the probability that<br />
the mean strength is found to be<br />
• between 24.9 and 25.1 kg?<br />
• less than 24.9 kg?<br />
EXERCISE 5. A wire company that manufactures wires <strong>for</strong> circus<br />
acts has taken a sample of 100 pieces of wire and wishes to<br />
see if the thickness of a batch of wires meets minimum<br />
specifications. Assume that the population data have a mean<br />
of μ = 0.45 cm with a standard deviation of σ= 0.03 cm.<br />
• Calculate the mean and standard error of the sampling<br />
distribution of means of such samples.<br />
• What may be said about the shape of this sampling<br />
distribution?<br />
• What is the probability that the sample mean is at least<br />
0.448 cm?<br />
EXERCISE 6. A certain trucking company wants to estimate the<br />
average tonnage of freight handled per month, and it has taken<br />
a sample of 36 months. Assume the true average tonnage per<br />
month is 225 tonnes, with SD of 30 tonnes. What is the<br />
probability that the sample mean will have a value within 7<br />
tonnes of the true mean?<br />
6<br />
<strong>Nan</strong> <strong>Carter</strong>: workshop notes <strong>prepared</strong> <strong>for</strong> <strong>Numeracy</strong> Centre Macquarie University.
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