STAT170 Workshop Notes prepared by Nan Carter for Numeracy ...

Week 6 H/Y. Questions on CONFIDENCE INTERVALS & HYPOTHESIS
TESTING prepared for Stat170 workshop held at the Numeracy
Centre, Macquarie University.
NOTE. FOR THIS TOPIC we still need to keep in mind that we
are working with two distributions:
• the distribution of the individuals in the population from
which we take a random sample; this distribution has mean μ
and standard deviation σ.
• the distribution of means of samples that are all the same
size (n); this is the sampling distribution of means of
samples of size n; this distribution has mean μ and standard
deviation σ/√n (also referred to as the ‘standard error’).
• If we need to use the sample standard deviation, s, in a
test of hypothesis about a mean μ, the variable is no longer
a z variable but a t-variable with n-1 degrees of freedom.
Similarly, in a 95% confidence interval the z-value 1.96 is
replaced by the appropriate t-value with (n-1) degrees of
freedom.
EXERCISE 1. The daily consumption of electric power in a
certain city is known to be normally distributed, and it is
claimed to have a mean of 6.5 units with a standard deviation
of σ = 1.5. In order to estimate the true mean daily power
consumption, the consumption was sampled on 18 randomly
selected days and the mean consumption was 6.973. (a) Test
the hypothesis that that daily consumption has not, on
average, changed. (b)Using the sample, find a 95% confidence
interval for the true mean daily power consumption, μ. How
confident are you that the claim that mean daily consumption
is 6.5 units is true?
EXERCISE 2. Washers are sold as having an inside diameter of 1.75cm. A sample of
200 washers produced by the machine has mean inside diameter of 1.77 cm. It is known
from past experience that the standard deviation of the diameter of washers produced
by this machine is 0.18 cm. (a) Test the hypothesis that the machine is still producing
washers with an inside diameter of 1.75 cm on average. (b) Using the sample, find a
95% confidence interval for the mean inside diameter of all washers currently being
produced by the machine. Are you at least 95% confident that the washers do on
average meet the claimed diameter size.
EXERCISE 3. Telecom wants to estimate the average length of telephone calls made
between two cities. Manager A claims it is close to 2 minutes, and Manager B disputes
this saying it is shorter, more likely 1.5 minutes. From a sample of 36 randomly selected
calls, it finds that the average length is 1.90 minutes. Historically, the standard deviation
of lengths of calls has been found to be 0.53 minutes.
• Using the data from the random sample, find a 95% confidence
interval for the true average length of calls. Which
Manager’s claim was closer? (A or B)
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Nan Carter: workshop notes prepared for Numeracy Centre Macquarie University.