STATS Module 9

• Hypothesis test


  A test of a certain given theory or belief about a population parameter
• Null hypothesis


  A claim (or statement) about a population parameter that is assumed to be true until it is declared false
• Alternative hypothesis


  A claim about a population parameter that will be declared true if the null hypothesis is declared to be false
• In a test of hypothesis, we test a certain given theory or belief about a population parameter
• We may want to find out, using some sample information, whether or not a given claim (or statement) about a population parameter is true

  
• This chapter discusses how to make tests of hypotheses about the population mean, μ, and the population proportion, p

  
• Two hypotheses


  • Null hypothesis
  • Alternative hypothesis
• Rejection and non-rejection regions

  Regions where the null hypothesis is rejected or not rejected

  
• Two types of errors

  • Type I error (rejecting a true null hypothesis)
  • Type II error (not rejecting a false null hypothesis)
• Tails of a test

  • Two-tailed test (rejection regions in both tails)
  • Left-tailed test (rejection region in left tail)
  • Right-tailed test (rejection region in right tail)
• A two-tailed test has rejection regions in both tails
• A left-tailed test has the rejection region in the left tail
• A right-tailed test has the rejection region in the right tail of the distribution curve

  
• The key word "changed" indicates a two-tailed test
• Two-tailed test


  H0: μ = $47,230 (The mean annual earning has not changed since 2014)

  H1: μ ≠ $47,230 (The mean annual earning has changed since 2014)
• If the alternative hypothesis has a not equal to (≠) sign, it is a two-tailed test

  
• The key phrase "less than" indicates a left-tailed test
• Left-tailed test


  H0: μ = or ≥ 12 ounces (The mean is equal to 12 ounces)

  H1: μ < 12 ounces (The mean is less than 12 ounces)
• When the alternative hypothesis has a less than (<) sign, the test is always left-tailed

  
• The key phrase "higher than" indicates a right-tailed test
• Right-tailed test


  H0: μ = or ≤ $471,257 (The current mean price is not higher than $471,257)

  H1: μ > $471,257 (The current mean price is higher than $471,257)
• When the alternative hypothesis has a greater than (>) sign, the test is always right-tailed
• Signs in H0 and H1 and tails of a test

  • Two-tailed test (H1: μ ≠ value)

  Left-tailed test (H1: μ < value)

  Right-tailed test (H1: μ > value)
• Two procedures to make tests of hypothesis

  • The p-value approach

  The critical-value approach

  
• 1. value approach


  Calculate the p-value for the observed value of the sample statistic, and compare it to the significance level to make a decision

  
• Critical-value approach


  Find the critical value(s) from a table and compare to the value of the test statistic to make a decision
• A left-tailed test has the rejection region in the left tail of the sampling distribution curve

  
• A right-tailed test has the rejection region in the right tail of the sampling distribution curve

  
• A two-tailed test has the rejection regions in both tails of the sampling distribution curve

  
• The p-value for a right-tailed test is the area in the upper tail of the sampling distribution curve to the right of the observed value

  
• The p-value for a left-tailed test is the area in the lower tail of the sampling distribution curve to the left of the observed value

  
• For a two-tailed test, the p-value is the sum of the areas in the two tails
• Steps to perform a test of hypothesis using the p-value approach


  State the null and alternative hypothesis

  2. Select the distribution to use

  3. Calculate the p-value

  4. Make a decision

  
• Hypothesis testing steps


  1. State the null and alternative hypothesis

  2. Select the distribution to use

  3. Calculate the p-value

  4. Make a decision

  
• Example 9-1 solution


  1. H0: μ = 90, H1: μ ≠ 90

  2. Use normal distribution since population standard deviation is known and sample size is small

  3. p-value = 2(.0007) = .0014

  4. Reject null hypothesis at α = .01 significance level, conclude mean time is different from 90 minutes

  
• Example 9-2 solution


  1. H0: μ ≥ 10, H1: μ < 10

  2. Use normal distribution since population standard deviation is known and sample size is large

  3. p-value = .0228

  4. Do not reject null hypothesis at α = .01, conclude mean weight loss is 10 pounds or more. Reject null hypothesis at α = .05, conclude mean weight loss is less than 10 pounds.
• In tests of hypotheses about μ using the normal distribution, the random variable (x-μ)/σ√(1/n) is called the test statistic

  
• Steps to perform a test of hypothesis with the critical-value approach

  1. State the null and alternative hypotheses

  2. Select the distribution to use

  3. Determine the rejection and nonrejection regions

  4. Calculate the value of the test statistic

  5. Make a decision

  
• Example 9-3 solution


  H0: μ = 12.44, H1: μ ≠ 12.44

  2. Use normal distribution since population standard deviation is known and sample size is large

  3. α = .02, two-tailed test with critical values -2.33 and 2.33

  4. Calculated test statistic z = 5.87

  5. Reject null hypothesis, conclude mean length is different from 12.44 minutes

  
• Example 9-4 solution


  H0: μ ≥ $300,000, H1: μ < $300,000

  2. Use normal distribution since population standard deviation is known and sample size is small but population is normal

  3. α = .025, left-tailed test with critical value -1.96

  4. Calculated test statistic z = -0.75

  5. Fail to reject null hypothesis, conclude mean net worth is at least $300,000