Confidence Intervals and Hypothesis Testing
Interpreting Results – sigma known
1) A research meteorologist has been studying wind patterns over the Pacific Ocean.
Based on these studies, a new route is proposed for commercial airlines going from San Francisco
to Honolulu. The new route is intended to take advantage of existing wind patterns to reduce flying
time.
Assume that flying times between the two cities are normally distributed with a mean of 5.25 hours
and standard deviation 0.6 hours. Thirty six flights on the new route have yielded a mean flying time
of 4.90 hours. Does this indicate that the average flying time for the new route is less than 5.25
hours?
a) Study the hypothesis testing results and write a conclusion within context. Does the conclusion
depend on a pre-established significance level? MUST EXPLAIN REASONING
The p-value = P(x-bar < 4.9) = 0.0002 is very low which means x-bar = 4.9
is inside the tail. It’s not very likely to observe a sample mean of 4.9 or lower
when the mean of the population is 5.25. Such a sample mean would be more
likely when mu is lower than 5.25.
We think that the
average flying times
for the new route are
lower than 5.25 hours.
We assume that the
mean flying time IS
EQUAL to 5.25 and
analyze how likely is it
to observe an x-bar of
4.9 when the
population mean is
5.25
What does the p-value mean: If we repeatedly selected samples of size 36
from a population with mu = 5.25, an x-bar of 4.9 or lower may be observed
in 2 out of 10,000 cases. Not a very likely outcome. This results suggests that
the mean of the population from which our sample was selected must have
been lower than 5.25
Such a small p-value gives a strong evidence against the null hypothesis (mu
= 5.25) and in favor of the alternative hypothesis (mu < 5.25)
Statistically significant results (at any of the common significance levels)- (z
= -3.5, p = 0.0002)- On average, flying times for the new rout are less than
5.25 hours.
b) A 99% confidence interval estimate for the mean flying time on the new route is shown. Does
the interval suggest that the average flying time for the new route is less than 5.25 hours?
MUST EXPLAIN REASONING
The interval provides plausible values for mu. Since all values in the interval
are lower than 5.25, the interval supports the claim that mu < 5.25 hours.
With 99% confidence we can say that on average, flying times for the new
rout are less than 5.25 hours.
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2) How does problem (1) change if the mean flying time of the thirty six flights selected on the new
route is 5.1 hours?
a) Study the hypothesis testing results and write a conclusion within context. Does the conclusion
depend on a pre-established significance level? MUST EXPLAIN REASONING
The p-value = P(x-bar < 5.1) = 0.067> 0.05 (this means that x-bar = 5.1 is
outside of the tail, close to 5.25)
At the 5% significance level (or any alpha lower than that) we don’t have
enough evidence to support the claim that mu < 5.25
Test results not statistically significant (z = -1.5, p = 0.067)
Note that, at the 10% significance level, we support the claim that mu <
5.25. (p = 0.067 < 0.10) In this case, the mean x-bar = 5.1 is in the tail
area.
In this case, the conclusion depends on the choice of the significance level
alpha.
b) A 90% confidence interval estimate for the mean flying time on the new route is shown. Does
the interval suggest that the average flying time for the new route is less than 5.25 hours?
MUST EXPLAIN REASONING
The interval contains the number 5.25.
Since the interval provides plausible values for mu, and 5.25 is in the
interval, it’s possible for mu to be 5.25. The interval does not support the
claim that in the new route the flying times are on average lower than 5.25
3) A machine makes twist-off caps for bottles. The machine is adjusted to make caps of diameter
1.85 cm. Production records show that when the machine is so adjusted, it will make caps with mean
diameters 1.85 cm and with standard deviation 0.05 cm. During production, an inspector checks the
diameters of caps to see if the machine has slipped out of adjustment. A random sample of 64 caps is
taken. If the mean diameter for this sample is 1.87 cm, does this indicate that the machine has
slipped out of adjustment and the average diameter of caps is no longer 1.85 cm? Use a 1%
significance level. Does the machine need an adjustment?
a) Study the hypothesis testing results and write a conclusion within context. Does the conclusion
depend on a pre-established significance level? MUST EXPLAIN REASONING
At the 1% significance level we support the claim that mu is not equal to
1.85, this means that the machine has slipped out of adjustment.
P = 2*P(x-bar > 1.87)= 0.001 < 0.01
Low probability value, x-bar = 1.87 is inside the tail – it’s unlikely to
observe such a sample mean, or a more extreme one, when mu = 1.85.
Statistically significant results at the 1% level. Z = 3.2, p = 0.0014
The conclusion will be the same at any of the common significance levels:
10%, 5%, 1%, 0.5%
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b) A 99% confidence interval estimate for the mean diameter of the twist-off caps is shown. Are
the results consistent with the hypothesis testing results? MUST EXPLAIN REASONING
The interval does not contain 1.81. With 99% confidence (1% significance
level) we can say that mu is not equal to 1.81.
Action has to be taken and the machine needs to be readjusted!
4) THINK ABOUT IT - Refer to problem (1)-b
Will a 95% or a 90% confidence interval estimate for the mean flying time on the new route provide
a DIFFERENT conclusion? Answer WITHOUT constructing the interval. EXPLAIN
REASONING.
A 95% or a 90% confidence interval will produce a shorter interval around x-bar of 4.9. All values
inside either interval will be lower than 5.25. Those intervals will also imply that mu < 5.25
5) THINK ABOUT IT - Refer to problem (1)
It’s not realistic to know the standard deviation of the population (sigma). Assume the given
standard deviation is the sample-standard deviation. Use the calculator to test the hypothesis and to
construct the 99% confidence interval. Show the results. Are the conclusions different to the ones
stated in parts 1-a and 1-b?
Let’s use s = 0.6 instead of sigma = 0.6
P value is a little larger than the one for the z-test, but it’s still lower than any of the common alpha
values. The conclusion is the same. Flying times along the new route are, on average, lower than 5.25
hours. (t = -3.5, p = 0.0006)
The interval is still completely below 5.25. The conclusion is the same.
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