01
Testing paperclips - Likelihood of error
A factory assembly line machine is cutting paperclips to length before folding. Each paperclip is supposed to be
long. The length of paperclips is approximately normally distributed with standard deviation . (a) Design a significance test with
that is based on the average of 5 measurements (sample mean). What is the rejection region? What is the probability of Type I error? (b) What is the probability of Type II error, given that the average paperclip length on the machine is actually
?
Solution
01
(a)
- Null Hypothesis
: “The expected paper clip length is not inches” - Alternative Hypothesis
: “The expected paper clip length is inches” Let
be the length of the paperclips for our sample. For the sample, assume and . Thus, . By symmetry, since we want a two-tailed test, it suffices to find the rejection region at one tail (we can then extrapolate for the second tail). We then have, for
the lower rejection region: Solving for
, we have . By symmetry, the lower bound for our upper rejection region is . Thus, our rejection region is By definition,
.
(b) A Type-II error occurs when the Null Hypothesis is incorrectly accepted, when it is actually false.
Thus,
. Thus, now let
. We then have: Link to original
02
Testing a coin by flipping until heads
Design a significance test to test the hypothesis that a given coin is fair. You think it may be biased towards tails.
Your test runs the following experiment: flip the coin repeatedly until the first time a heads comes up. Let
be the flip number of the first heads. This is your decision statistic. Your test should have significance level
. Which of these coins would pass your test?
- Two-headed coin
- Two-tailed coin
- Both
- Neither
03
Valves at various temperatures
The lifetime of a certain fuel injection valve is known to follow an exponential distribution,
, where in failings per year and is the ambient temperature in degrees Celsius. Sometimes the valves fail a good deal more frequently than usual, possibly due to cracked gaskets used in construction. To detect failings from cracked gaskets, each day the following test is performed:
valves are monitored in use at for the full day and the number that fail is recorded. (a) Suppose a significance test is designed such that it rejects the hypothesis “normal valves, no cracked gaskets” when just one (or more) fail the test. What is the significance level of this test, as a function of
? (b) How many valves would have to be tested at
in order to achieve a significance of ? (Find using the function resulting from (a).) (c) Is
(to achieve ) increasing, decreasing, or constant with increasing test temperature?
04
Blue eyes
A redditor claims that 10% of people have blue eyes, but you think it is not that many. You work at the DMV for the summer, so you write down the eye color recorded on drivers’ licenses of various people in the database.
(a) Suppose you record the eye color of 1000 people and let
be the number that are blue. If the rejection region is , what is the significance level of the test? (b) Take again the experiment in (a). If you want a significance level of
, what should the rejection region be in your test? (c) Suppose the fact is that 7% of people have blue eyes. How likely is it that your test in (b) rejects
?
05
Shipping time test
The number of days it takes for a package to arrive after being shipped with a particular company is a random variable,
. When the shipping process is operating at full capacity and delays are not common, the PMF of is given in the following table:
1 2 3 4 5 6 7 8 9 0.041 0.229 0.379 0.237 0.045 0.021 0.019 0.017 0.012 Design a significance test at the
level that uses the value of X for one package to test the null hypothesis: the shipping process is operating at full capacity. You should clearly state which values of X are in the rejection region.
Solution
03
The rejection region is
Link to original. So if shipping takes 8 or 9 days, we will reject .
06
Bits received in error
In a digital communication channel, it is assumed that a bit is received in error with probability
. Someone challenges this hypothesis: they believe the error rate is higher than . Assume 100,000 bits are transmitted. Design a one-tailed significance test using and , the number of bits received in error, to decide whether to reject the hypothesis that the error rate is . Your rejection region should be of the form . You do not have to use the continuity correction.
Solution