1. Suppose you are in charge of inventory maintenance at a bicycle shop. One of
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1. Suppose you are in charge of inventory maintenance at a bicycle shop. One of your jobs is to ensure that the tire pressure in each of the display bicycles is between 65-85 PSI (pounds per square inch). If the pressure is too low, then there is a risk of wheel damage when a customer rides out on one. On the other end, if the pressure is too high, there is a (small) risk of the tire exploding! Of course, you don't know the true pressure of any particular tire. Instead, you have the output from your pressure gauge. true pressure, but not necessarily the same. Laboratory testing of the particular gauge you use has shown that there is a ±2 PSI error margin, and so to be careful, you decide that you will adjust the pressure on any tire that has a measured pressure above 81 PSI or below 69 PSI, giving you 2x the error margin on either side. Previous testing shows that this procedure will give the following results: This will be similar to the Measure inside 69-81PSI | Measure outside 69-81 PSI PSI within 65-85 PSI5000 PSI outside 65-85 PSI 105 (e) What is the rejection region for the test you are conducting? (f) Practically speaking, which error seems more problematic: a Type I error or a Type II error? (g) If we wanted to decrease the Type I error rate, should we increase or decrease the rejection region? (h) If we wanted to decrease the Type II error rate, should we increase or decrease the rejection region?Explanation / Answer
Here the null hypothesis H0 : Gauge is measuring the pressure correctly with 2x margin
and alternate hypothesis H1 : Gauge is not measuring the pressure correctly with 2x margin
e. Hence the rejection region is as follows:
1. PSI within 65-85 and measure outside 69-81 PSI
2. PSI outside 65-85 and measure inside 69-81 PSI
f. Both Type 1 and Type 2 error are problematic in general and which one is more worse very much depends upon th scenario and the severiy that is associated with it.
For example : If a medical test for cancer gives false results then, Type 1 error would lead to a person going ahead with further test wherein it will be confiremed that he is not suffering from cancer. However, the Type 2 error will lead a person to believe that he is fine wheras he is actually not. Hence Type 2 is more vulnerable.
If we take an example of a person being suspected of crime then, in that case Type 1 error is more problematic where a person who has not comiited the crime is being punished.
g. To decrease the type 1 error, we need to decrease the rejection region. This is because if we have a look at the graph for normally distributed data increase in rejection region reduces the non rejection region leading to increased probability of rejecting the hypothesis which is true.
h. To decrease the type 2 error, we need to increase the rejection region. This is because if we have a look at the graph for normally distributed data increase in rejection region results in reduction of non rejection area, thus, decreasing the probability of accepting the hypothesis which is not true.
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