A city police department is considering replacing thetires on its cars with a new brand tires. If μ1 is the average number of miles that the old tires last and μ2 is the aver-age number of miles that the new tires will last, the null hypothesis to be tested is μ1 = μ2.(a) What alternative hypothesis should the departmentuse if it does not want to use the new tires unless theyare definitely proved to give better mileage? In otherwords, the burden of proof is put on the new tires, andthe old tires are to be kept unless the null hypothesis canbe rejected.(b) What alternative hypothesis should the departmentuse if it is anxious to get the new tires unless they actuallygive poorer mileage than the old tires? Note that now theburden of proof is on the old tires, which will be kept onlyif the null hypothesis can be rejected.(c) What alternative hypothesis should the departmentuse so that rejection of the null hypothesis can leadeither to keeping the old tires or to buying the newones?
Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
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Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
A city police department is considering replacing the
tires on its cars with a new brand tires. If μ1 is the average
number of miles that the old tires last and μ2 is the aver-
age number of miles that the new tires will last, the null
hypothesis to be tested is μ1 = μ2.
(a) What alternative hypothesis should the department
use if it does not want to use the new tires unless they
are definitely proved to give better mileage? In other
words, the burden of proof is put on the new tires, and
the old tires are to be kept unless the null hypothesis can
be rejected.
(b) What alternative hypothesis should the department
use if it is anxious to get the new tires unless they actually
give poorer mileage than the old tires? Note that now the
burden of proof is on the old tires, which will be kept only
if the null hypothesis can be rejected.
(c) What alternative hypothesis should the department
use so that rejection of the null hypothesis can lead
either to keeping the old tires or to buying the new
ones?
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