Understanding Statistical Hypothesis Testing: A Comprehensive Guide, Slides of Data Analysis & Statistical Methods

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9: Tests of

Hypotheses

EM 7: Engineering Data Analysis Second Semester, 2019- Pamantasan ng Lungsod ng Valenzuela

Introduction

• Often, the problem confronting a researcher is not so much of the estimation of a population parameter, as discussed previously, but rather the formation of a data-based decision procedure that can produce a conclusion about some scientific system.
• In cases like these, the researcher postulates or conjectures something about a system. The conjecture can be put in the form of a statistical hypothesis, which is a major area of statistical inference. 2

Introduction

• It should be clear that the decision procedure must include an awareness of the probability of a wrong conclusion.
• The analyst should be accustomed to understanding that rejection of a hypothesis implies that the sample evidence refutes it. In other words, rejection means that there is a small probability of obtaining the sample information observed when, in fact, the hypothesis is true. 4

Introduction

• Example: for a sample of 100, if 20 were found to be defective, it is certainly an evidence for rejection if we have the condition of a binomial parameter 𝑝 = 0.10. If 𝑝 = 0.10, the probability of obtaining 20 or more defectives is approximately 0.002. With the resulting small risk of a wrong conclusion, it would be safe to reject the hypothesis that 𝑝 = 0.10.
• In other words, rejection of a hypothesis tends to all but “rule out” the hypothesis. On the other hand, it is very important to emphasize the acceptance or failure to reject does not rule out other possibilities. 5

Null and Alternative

Hypothesis

• The structure of hypothesis testing will be formulated through the use of null hypothesis 𝐻଴ which refers to any hypothesis we wish to test.
• The rejection of 𝐻଴ leads to the acceptance of an alternative hypothesis, or 𝐻ଵ.
• The alternative hypothesis 𝐻ଵ usually represents the question to be answered or the theory to be tested, and thus, its specification is crucial.
• The null hypothesis 𝐻଴ nullifies or opposes 𝐻଴ and is often the logical complement to 𝐻ଵ. 7

Null and Alternative

Hypothesis

• Please do take note that the analyst arrives at one of the two following conclusions: - Reject 𝑯𝟎 in favor of 𝐻ଵ because of sufficient evidence in the data; or, - Fail to reject 𝑯𝟎 because of insufficient evidence in the data
• Note that the conclusions do not involve a formal and literal “accept 𝐻଴”. The statement of 𝐻଴ often represents the status quo in opposition to the new idea, conjecture, and so on, stated in 𝐻ଵ, while failure to reject 𝐻଴ represents the proper conclusion. 8

Testing a Statistical

Hypothesis

10

Testing a Statistical

Hypothesis

Example: A certain type of cold vaccine is known to be only 25% effective (𝑝 = 0.25) after a period of 2 years. To determine if a new and somewhat more expensive vaccine is superior in providing protection against the same virus for a longer period of time, suppose that 20 people are chosen at random and vaccinated. If more than 8 (randomly decided, but somewhat reasonable) of those receiving the new vaccine surpass the 2-year period without contracting the virus, the new vaccine will be considered superior to the one presently in use. 11

Testing a Statistical

Hypothesis

The test statistic on which we base our decision is X, which is the number of individuals who receive protection from the new vaccine for a period of at least 2 years. Possible values of X from 0-20, are divided into two groups: those numbers less than or equal to 8 and those greater than 8. 13

Testing a Statistical

Hypothesis

All possible scores greater than 8 constitute the critical region. The last number that we observe in passing into critical region (also 8) is called the critical value. Therefore, if 𝑥 > 8, we reject 𝐻଴ in favor of the alternative hypothesis 𝐻ଵ. If 𝑥 ≤ 8, we fail to reject 𝐻଴. 14

Type I and Type II Errors

A second error is committed if 8 or fewer of the group surpass the 2-year period successfully and we are unable to conclude that the vaccine is better when it actually is better (𝐻ଵ true). In this case, we fail to reject 𝐻଴ when in fact, 𝐻଴ is false. This is called a type II error. 16 DEFINITION: Non-rejection of the null hypothesis when it is false is called a type II error.

Type I and Type II Errors

In testing any statistical hypothesis, there are four possible situations that determine whether our decision is correct, or in error. The probability of committing a type I error, is called the level of significance, 𝛼. 17 𝐻଴ is true 𝐻଴ is false Do not reject 𝐻଴ Correct decision Type II error Reject 𝐻଴ Type I error Correct decision

Type I and Type II Errors

The probability of committing a type II error (𝛽) is impossible to compute unless we have a specific alternative hypothesis. For example, if we test the null hypothesis that 𝑝 = 1/4 against the alternative hypothesis that 𝑝 = 1/2, then we can compute the probability of not rejecting 𝐻଴ when it is false. That is the probability of obtaining 8 or fewer in the group that surpass the 2-year period when 𝑝 = 1/2. 𝛽 = 𝑃 𝑡𝑦𝑝𝑒 𝐼𝐼 𝑒𝑟𝑟𝑜𝑟 = 𝑃 𝑋 ≤ 8 𝑤ℎ𝑒𝑛 𝑝 = 1 2 = ෍ 𝑏 𝑥; 20 , 1 2 ଼ ௫ୀ଴ = 0. 2517 19

Type I and Type II Errors

The value 0.2517 is a rather high probability, indicating a test procedure in which it is quite likely that we shall reject the new vaccine when, in fact, it is superior to what is now in use. If the testing director is willing to make a type II error if the more expensive vaccine is not significantly superior (the only time he wishes to guard against the type II error is when the true value of 𝑝 is at least 0.70): 𝛽 = 𝑃 𝑡𝑦𝑝𝑒 𝐼𝐼 𝑒𝑟𝑟𝑜𝑟 = 𝑃 𝑋 ≤ 8 𝑤ℎ𝑒𝑛 𝑝 = 1 2 = ෍ 𝑏 𝑥; 20 , 0. 7 ଼ ௫ୀ଴ = 0. 0051 20 It is then extremely unlikely that the new vaccine would be rejected when it was 70% effective after a period of 2 years.