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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• SO: if we have only one sample and find its mean, ybar, we can say that we are 95%<br />
confident that the true mean μ lies within the interval (ybar - 1.96 x σ/√n to ybar + 1.96 x<br />
σ/√n). This is what is called a 95% confidence interval <strong>for</strong> the population mean μ.<br />
EXERCISE 1. The daily consumption of electric power in a certain city is known to be<br />
normally distributed with a standard deviation of σ = 1.5. In order to estimate the true<br />
mean daily power consumption, the consumption was sampled on 18 randomly selected days<br />
and the mean consumption was 6.973. Find a 95% confidence interval <strong>for</strong> the true mean<br />
power consumption, μ.<br />
EXERCISE 2. The mean inside diameter of a sample of 200<br />
washers produced <strong>by</strong> a machine is 1.77 cm. It is known from<br />
past experience that the standard deviation of the diameter of<br />
washers produced <strong>by</strong> this machine is 0.18 cm. Find a 95%<br />
confidence interval <strong>for</strong> the mean inside diameter of all<br />
washers produced <strong>by</strong> the machine.<br />
EXERCISE 3. Telecom wants to estimate the average length of<br />
telephone calls made between two cities. From a sample of 36<br />
randomly selected calls, it finds that the average length is<br />
1.90 minutes. Historically, the standard deviation of lengths<br />
of calls has been found to be 0.53 minutes.<br />
• Find a 95% confidence interval <strong>for</strong> the true mean length of calls.<br />
• (optional) What would be a 99% confidence interval?<br />
EXERCISE 4. Suppose we wanted to place an order <strong>for</strong> wire that had a breaking strength of<br />
8kg on average with a standard deviation of 1.6kg. We first bought a random sample of 50<br />
lengths, measured the breaking strength of each and found the mean was 7.8kg. Could we be<br />
95% confident that a large order would meet our specification on average?<br />
EXERCISE 5. A quality control engineer wishes to check the mean weight of potato chip<br />
bags produced on his machines. He takes a random sample of 36 bags and find their mean<br />
weight is 198 grams. Assuming that the standard deviation of all weights is σ = 3.6 grams,<br />
Can he be 95% sure that the machines are producing bags that are on average 198.8 grams?<br />
EXERCISE 6. A metallurgist claims that the mean hardness of die-cast aluminium is 13.7<br />
with a standard deviation of 0.8. He gives us a random sample of 50 pieces and we find the<br />
mean hardness is 14.3. We claim these are likely to be too hard <strong>for</strong> our purposes. Are we<br />
making the correct decision?<br />
EXERCISE 7. Go back to Q1, and use a test of hypothesis to answer the research question: is<br />
the consumption of power equal to 6.5?<br />
EXERCISE 8: In Q2 the research question is whether the mean inside diameter is on average<br />
1.80cm.<br />
EXERCISE 9: In Q3 test the hypothesis that the mean length of call is 2 minutes.<br />
EXERCISE 10: In Q4 use a hypothesis test to find if a large order would meet our<br />
specification on average.<br />
16<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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