~Class Meeting  #20-Hypothesis Testing  (preliminaries)

~Note:  Rare Event Rule: (common sense): If the probability of an event is extremely small, under a given assumption, then we conclude that the assumption is probably incorrect. (rolling a 7 with a pair of dice 30 out of 35 times, under the assumption the dice are fair (well balanced), has an extremely small probability. So, we would conclude that the dice were loaded (off balance in favor of a 7)

~Note:  If we assume a certain make of car gets 26 mpg and, from a sample of 50 of those cars, find that that they get 16 mpg on average, We would probably conclude that the original assumption of 26 mpg is incorrect.

~Terms & definitions

~Test statistic: A value computed from the sample data and is used to make a decision about a given assumption about the population.

Since the population parameters we will be testing in this unit are the population proportion, p, and the population mean, m, the test statistics would involve the sample proportion, p', and the sample mean x'. The z scores or t scores can be calculated by the following formulas that were previously covered. (x=the number (must be an integer) in the sample containing the given attribute)

z=(x-m)/s =(x-np)/sqr(npq)=[(x/n)-p]/sqr(pq/n)=(p' - p)/sqr(pq/n) for population proportions

z=(x' - m)/[s/sqr(n)],       t = (x'-m)/[s/sqr(n)], for population means. s is the standard deviation from our sample.

~Note: Use the value of p or m in the null hypothesis when computing the Test Statistic.
     
                               

~Hypothesis:  a statement that something is true.

~Null Hypothesis:  a hypothesis to be tested. Symbol given is Ho  (statement that the population value is EQUAL to some claimed value).

~Alternate Hypothesis:  a hypothesis to be considered as an alternate to Ho. Symbol given is H₁ or H_a. (statement that the population value differs from Ho) (<, >, or not equal to the value stated in Ho).

~Decision Rule:  using the test statistic to specify whether or not Ho should be rejected in favor of H₁.

~Critical Regions:   areas of rejection for the null hypothesis

~Significance Level:  (denoted by alpha). The probability that the test statistic falls in the critical region when Ho is actually true.

~If the test statistic falls in the critical region, then we reject Ho. Popular choices for alpha are .05, .01, & .10.

~Critical Value:  value(s) at the boundary of the critical regions  (separates rejected & non rejected regions).

~P - Value:  (probability value). The Area to the right of the test statistic (right-tailed test), left of it (left-tailed test), or both (2-tailed test).

~If a p-value is less than the significance level alpha, we reject Ho. Otherwise, we do not reject Ho.

~Type I Error:  rejecting a true null hypothesis.

~Note: alpha is the probability of making a type I error.
                                                                         
~Type II Error:  not rejecting a false null hypothesis.

~Note: Beta is the probability of making a type II error
             (see Topics of Interest on how to find beta)    

~Two-tailed test: Use when H₁ does not equal the value in Ho & no direction is indicated or assumed.

~Left-tailed test:  Use when H₁ < the value in Ho.

~Right-tailed test:  Use when H₁ > the value in Ho.

~Note:  If we do not reject Ho, we conclude only that the data did not provide sufficient evidence to support H₁. We do not conclude that the data provided sufficient evidence to support Ho. If Ho is rejected, we conclude the H₁ is probably true.

See Hypothesis Testing for more detail