STAT 101 Chapter 9: Hypothesis Testing

• Introduction to Probability and Statistics, STAT 101 covers Large-Sample Tests of Hypotheses in Chapter 9
• The chapter is presented by Dr. Abdel-Salam G. Abdel-Salam from the Department of Mathematics, Statistics and Physics at Qatar University

Learning Objectives

• Learn about Hypothesis Testing (HT)
• Understand the relationship between Hypothesis Testing and Confidence Intervals (CI)
• Learn the factors affecting Confidence Intervals
• Learn how to calculate sample size
• Learn about the types of errors in hypothesis testing
• Learn about the power of a test
• Chapter 9 covers sections 9.1, 9.2, and 9.3

Outline of Chapter 9

• Introduction to Hypothesis Testing
• Sample Size Calculation for Estimating Population Mean
• Decisions and Types of Errors

Introduction to Hypothesis Testing

• Hypothesis testing (HT) examines a claim about a parameter's value
• The HT procedure consists of basic steps followed rigorously
• The steps play an important role in the HT process
• The order of steps is important as some require information from prior steps
• Repetition familiarizes you with the procedure

Null vs Alternative Hypothesis

• The null hypothesis (Ho) states a specific value for the parameter
• The alternative hypothesis (Ha) suggests the parameter falls within an alternative range of values

Steps for Significance Test of Population Mean (μ)

• State the practical question requiring a statistical test, identifying the variable and parameter
• Formulate Hypotheses:
  • Null Hypothesis (H0): μ = μ0, where μ0 is the hypothesized value
  • Alternative Hypothesis (Ha): μ ≠ μ0 (two-sided test) or μ < μ0 or μ > μ0 (one-sided)
• Solve by checking test conditions, ensuring data is randomized, and confirming population distribution is N(μ, σ) with σ known
• Calculate the test statistic: z = (x̄ - μ0) / (σ / √n)
• Determine the P-value, using a table to find tail probabilities based on the alternative hypothesis

Steps for Significance Test of Population Mean (μ) Continued

• State the practical question and define the variable and parameter
• Formulate Hypotheses
• Solve
• Determine the P-value
• Conclusion: Smaller P-values are stronger evidence against H0
• needed, reject H0 if the P-value is less than or equal to the preselected significance level α (e.g., 0.05)
• Relate the conclusion to the context of the study

Example Test About Means

• An insurance company claims the mean medical expense is at least $700 per year for American families:
  • Survey of 30 families found expense was $640
  • Test the claim at α = 0.05, knowing σ = 140
• State: Claim states the mean medical expense is at least $700
  • The sample mean of 30 families is $640
  • Determine evidence if true mean medical expense is lower than the company’s claim
• Hypotheses:
  • H0: μ ≥ 700 vs Ha: μ < 700 (one-sided test), where μ is the true mean

Example Test About Means Continued

• Hypotheses: H0: μ ≥ 700 vs. Ha: μ < 700 (one-sided test)
• Check conditions: Assume medical expenses follow a normal distribution with σ = 140
• Test statistic: z = (640 - 700) / (140 / √30) = -2.347
• P-value use a table for Ha: μ < 700 to find the left-tail probability
• P-value = P(z < -2.347) ≈ 0.0094
• Conclusion: With p = 0.0094, sufficient evidence exists to reject the insurance company's claim average expense is at least $700, indicating the true average expense may be statistically significantly less

Tests from Confidence Interval

• A continued example asks if results would differ if α = 0.005
• In certain cases, there is a connection between HT and CI
• For two-sided tests of μ, HT and CI results are identical for a fixed α
• Essentially, the CI is the non-rejection region and is the complement to the rejection region for the HT
• Statistical and practical significance differ, and neither implies the other
• "Fail to reject" does not mean "accept"

Additional Example

• Homeowner samples 64 homes finding the average selling price is $252,000 with a standard deviation of $15,000:
  • Determine if the evidence concludes that the average selling price is greater than $250,000
  • Use a = .01.
• H0: μ = 250,000 vs. Ha: μ > 250,000
• Test statistic: z = (252,000 - 250,000) / (15,000 / √64) = 1.07
• Rejection Region: Reject H0 if z > 2.33; the test statistic falls in the rejection region, and the p-value is less than a = .01.
• z = 1.07 does not fall in the rejection region and H0 is not rejected
• There is not enough evidence to indicate that μ is greater than $250,000

Sample Size Calculation

• Key results for finding the sample size:
  • The margin of error (m) depends on the standard error (se)
  • The standard error (se) depends on the sample size
• A confidence interval for the mean of a Normal population has a specified margin of error m when the sample size is n = (z*σ / m)^2
  • z is the z-score corresponding to the desired confidence level
  • σ is the population standard deviation
  • m is the desired margin of error

Sample Size Calculation Example and Solution

• In a previous example, the data in the doctor's record has σ = 17.7, so it is used as our guess
• The formula:
  • n = (z*σ / m)^2
• Solution is sample size is about 34 to necessarily guarantee a margin of error at most 6

Decisions and Types of Errors

• Procedures aren't always correct 100% and have two possible errors that can occur:
  • Type I Error: Rejecting H0 when H0 is true
  • Type II Error: Failing to reject H0 when H0 is false
• Based on this is the Truth Table
  • If you fail to reject Ho when Ho is true this is a correct decision
  • If you reject Ho when Ho is true this is a Type I Error
  • If you fail to reject Ho when Ha is true This is a Type II Error
  • If you reject Ho when Ha is true this is a correct decision

Probabilities of Outcomes

• P(Fail to reject H0 | H0 true) = 1 - α = Confidence level
• P(Reject H0 | H0 true) = P(Type I error) = α = Significance level
• P(Fail to reject H0 | Ha true) = P(Type II error) = β
• P(Reject H0 | Ha true) = 1 - β = Power
• The Power of a HT detects a difference when one actually exists

Comments on Hypothesis Testing

• When conducting HT, the type I error rate, α, is set, but do nothing about the type II error rate, β- In general, this is the case:
  • When α is fixed, the value of β can be determined for any value of the parameter under the alternative hypothesis
  • As α increases, β decreases
  • This is somewhat like the trade-off between validity and precision with CI
• It is important that when doing HT in the real world to set up the hypotheses so that the type I error is what we are more worried about, since we can control the rate at which they occur

Testing for Water Pollution Example

• Testing for water pollution where:
  • H0: μ ≤ μ0 vs. Ha: μ > μ0
• μ is the average amount of pollutant measured as ppm and μ0 is the threshold for safe water
  • Type I error: Reject H0 when H0 is true -Declare water unsafe when safe
  • Type II error: Fail to reject H0 when Ha is true -Declare water safe when unsafe
  • Type II error is worse
• What to do: test H0: μ ≥ μ0 vs. Ha: μ < μ0
  • Control the more important of the two types of errors

Summary of Key Points

• Test statistics are of the form: (value of statistic - hypothesized value of parameter) / standard error of statistic
• State the formula first and substitute in the values when running HT and CI
• Differentiate between zobs, z1-α/2, and Z:
  • The first is an observed test statistic
  • The second is a critical value
  • The third is a random variable
• A connection exists between HT and CI

Summary Continued

• Practical and statistical significance differ; neither implies the other
• “Fail to reject” differs from "accept," like "not guilty" differs from "innocent"
• Reporting the p-value is preferred over the critical value method
  • It allows other researchers to look at the results and evaluate them at their own significance level and it provides more widely acceptable results
• Power: The end of the chapter has more on how to compute the probability of a Type II error, and how to compute power
  • Power computations are done assuming the alternative hypothesis is true
    • This can be more complicated since the alternative can be true in many different ways (is μ₁ = 10 or is μ₁ = 20?)