STATISTICS FOR MANAGERS

B. Hypothesis Testing - One Sample

1. Basic concepts

a. Null hypothesis (H₀) - theory, claim, or assertion about a population parameter

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b. Alternative hypothesis (H₁) - conclusion if H₀ found to be false

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• Hypotheses are about the population parameter, not the sample statistic

• Null hypothesis always contains an equality sign

• Alternative hypothesis never contains an equality sign

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c. Test statistic - a value based on sample results that is used to determine the likelihood of the H₀ being true

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d. Region of rejection - reject H₀ if test statistic falls in this region

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e. Region of nonrejection - do not reject H₀ if test statistic falls in this region

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f. Critical value - divides region of rejection from region of nonrejection

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g. Type I error - rejecting H₀ when H₀ is true

h. Type II error - not rejecting H₀ when H₀ is false

i. Level of significance (α) - probability of a Type I error

j. Confidence coefficient (1 - α) - probability that H₀ is not rejected when H₀ is true

k. Beta risk (β) - probability of a Type II error

l. Power of a test (1 - β) - probability of rejecting H₀ when H₀ is false

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Tradeoff between Type I and Type II errors:

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2. Z test for the population mean, population standard deviation known

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a. Procedure

(1) State H₀ and H₁

(2) Choose level of significance and sample size

(3) Determine appropriate test statistic and sampling distribution

(4) Determine critical value

(5) Collect data and compute value of test statistic

(6) Make appropriate decision

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b.  Relationship to confidence intervals:

• If population parameter is outside confidence interval, H₀ will be rejected

• If population parameter is inside confidence interval, H₀ will not be rejected

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c. p-value approach

• p-value - probability of obtaining a test statistic equal to or more extreme than the result obtained from the sample data, given that the null hypothesis is true

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d. One-tailed tests

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3. t test for the population mean, population standard deviation unknown

• Use sample standard deviation to approximate population standard deviation => t-distribution is appropriate

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4. Z test for the population proportion

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5. Use of Excel and PHStat

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6. Ethical considerations

• Data collection issues - randomization, informed consent

• Two-tailed vs one-tailed tests

• Choice of level of significance

• Data snooping

• Data cleansing

• Reporting of findings