BA Chapter 7a

Question 1

• Definition: Involves drawing inferences about 2 contrasting propositions (each called a hypothesis) relating to the value of one or more population parameters.
• H₀ Null hypothesis: describes an existing theory

H₁ Alternative hypothesis: the complement of H₀

• Using sample data, we either:
  
  • reject H₀ and conclude H₁, or
  • fail to reject H₀ and conclude that there is not enough evidence to reject H₀.
• NOTE: we NEVER say that we Accept H₀!!! We either reject or fail to reject H₀.

Answer 1

• An Hypothesis Test evaluates a claim made about the value of a population parameter by using a sample statistic.
• Steps of hypothesis testing procedures:
  
  • Step 1: Identify the population parameter and formulate the hypotheses to test.
    
    • NOTE: the equality (>=, <=, or =) ALWAYS goes in the null hypothesis (H₀).
    • The alternative (H₁) is the opposite (<, >, ≠). For example: H₀: µ = 25; H₁: µ ≠ 25
  • Step 2: Select a level of significance α (alpha). This defines the risk (probability) of rejecting H₀ when in fact it is true. For example, α = 0.1, α = 0.05, or α = 0.01
  • Step 3: Calculate the appropriate Test Statistic and compare to the sampling distribution’s Critical Value from a table. (Example of sampling distributions: Z, t, F, X², etc.).
  • Step 4: Either reject or fail to reject H₀ based on decision rule.
  • Step 5: Draw a conclusion.

Question 2

• Definition: a hypothesis test that involves a only one population parameter. 3 forms:
  
  • H₀: population parameter >= constant vs. H₁: population < constant
  • H₀: population parameter <= constant vs. H₁: population parameter > constant
  • H₀: population parameter = constant vs. H₁: population ≠ constant
• NOTE: The equality part of the hypotheses is always in the Null hypothesis.

Answer 2

• How to determine the proper form of the null and alternative hypotheses?
  
  • Hypothesis testing always assumes that H₀ is true and uses sample data to determine whether H₁ is more likely to be true.
  • We cannot prove that H₀ is true; we can only fail to reject it. Thus, if we cannot reject the null hypothesis, we have shown only that there is insufficient evidence to conclude that the alternative hypothesis is true.
  • However, rejecting the null hypothesis provides strong evidence that the null hypothesis is not true and the alternative hypothesis is true. Thus, what we wish to prove should be identified as the alternative hypothesis.

Question 3

Hypothesis Testing can generate 1 of 4 outcomes:

• The null hypothesis is actually true, and the test fails to reject it.
  
  • the probability that P(not rejecting H₀ l H₀ is true) is called the Confidence Coefficient, calculated as 1 - α. For a confidence coefficient of 0.95, means that we expect 95 out of 100 samples to support the null hypothesis rather than the alternate hypothesis.
• The null hypothesis is actually false, and the hypothesis test rejects it.
• The null hypothesis is actually true, but the hypothesis test rejects it (Type I error).
  
  • the probability of making a Type I error: P(rejecting H₀ l H₀ is true) is denoted by α, which is called the level of significance. Commonly used levels for α are 0.1, 0.05, and 0.01.
• The null hypothesis is actually false, but the hypothesis test fails to reject it (Type II error).
  
  • we cannot control the probability of a Type II error, P (not rejecting H₀ l H₀ is false), which is denoted by β. Unlike α, β cannot be specified in advance but depends on the true value of the unknown population parameter.
  • 1 - β is called the power of the test and represents the probability that P (rejecting H₀ l H₀ is false). The power of the test is sensitive to the sample size; 1 - β can be increased by taking larger samples.

Answer 3

Selecting the test statistic:

• The decision to reject or fail to reject a null hypothesis is based on computing a Test Statistic (TS).
• Different types of hypothesis tests use different TS, and it’s important to use the correct one. Proper TS often depends on whether or not the std deviation is know.
• 2 types of one-sample hypothesis tests:
  
  • one-sample test for mean, σ known - formula is on page 203
  • one-sample test for mean, σ unknown - formula is on page 203

Question 4

Drawing a conclusion:

• The decision to reject or fail to reject null hypothesis is based on comparing the value of the TS to a Critical Value (CV).
• The CV divides the sampling distribution into 2 parts, a rejection region and a nonrejection region. If the TS fall into the rejection region, we reject the null hypothesis; if the TS fall outside of the rejection region, we fail to reject it.

(continued on the back)

Answer 4

• When the null hypothesis is structured as = and the alternative hypothesis as ≠, the rejection region will occur in both the upper and lower tail of the distribution. This is called a two-tailed test of hypothesis. The combined area of both tails must be α; each tail has an area of α/2.
• One-tailed tests of hypothesis. The rejection region occurs only in one tail of the distribution. If H₁ is stated as <, the rejection region is in the lower tail; if H₁ is stated as >, the rejection region is in the upper tail.
• NOTE: two-tailed tests have BOTH upper and lower CV; one-tailed tests have either a lower or upper CV; for std normal and t-distributions, which have a mean of 0, lower-tail CV is negative, upper-tail CV is positive.

Question 5

Two-Sample Hypothesis Test: test for differences in two population parameters.

• Forms:
  
  • Lower-tailed test H₀: population parameter 1 - population parameter 2 >= 0 vs. H₁: population parameter 1 - population parameter 2 < 0. This test seeks evidence that population parameter 2 > parameter 1.
  • Upper-tailed test