statistics

• Hypothesis Testing


  A process wherein we make decisions in evaluating claims about the population based on the characteristics of a sample taken from the same population
• Steps of Hypothesis Testing


  1. State the null and alternative hypotheses

  2. Select the level of significance

  3. Statistical Tool (Z-test, T-Test, CLT)

  4. Formulate the decision rule

  5. Compute the value of the test statistics

  6. Make a Decision

  
• Null hypothesis


  A statement about the value of a population parameter formulated with the hope of it being rejected. It is usually denoted by Ho.

  
• Alternative hypothesis


  If Ho is rejected, we will be led to accept an alternative hypothesis, usually denoted by Ha.
• A null hypothesis always involves an equality symbol. ( =, ≥, ≤ )
• An alternative hypothesis contains the inequality symbol. ( >, <, ≠ )
• Example 1: Hypothesis testing for class size

  • Ho: The mean class size in public schools is equal to 65. Ho: μ ≥ 65 or Ho: μ = 65

  Ha: The mean class size in public schools is less than 65. Ha: μ < 65
• Example 2: Hypothesis testing for multivitamin turnover rate


  • Ho: The mean turnover rate of a 200-tablet bottle of multivitamins is equal to 6.0. Ho: μ = 6.0

  Ha: The mean turnover rate of a 200-tablet bottle of multivitamins is no longer 6.0. Ha: μ ≠ 6.0
• Level of significance (α)

  The probability of rejecting the null hypothesis when it is true

  
• The smaller value of α, the surer we are that we are not making an error if we end up rejecting Ho.
• There is no fixed value of α in any hypothesis test. While α = 0.05 tends to be chosen as a default value, the choice may differ depending on the application.

  
• Usually, α is selected at 0.05 for consumer research projects, 0.01 for quality assurance, and 0.10 for political polling.
• The usual significance level in research or social science research is 5% (α = 0.05). It means that there is a 5% chance of rejecting a true null hypothesis and we are 95% confident that the result is true.
• Type I Error


  Rejecting the null hypothesis when it is true

  
• Type II Error


  Failing to reject the null hypothesis when it is false

  
• If Donna finds out that her null hypothesis is true and she fails to reject it, then she commits a Correct Decision.
• If Donna finds out that her null hypothesis is true and she rejects it, then she commits a Type I Error.
• If Donna finds out that her null hypothesis is false and she fails to reject it, then she commits a Type II Error.
• If Donna finds out that her null hypothesis is false and she rejects it, then she commits a Correct Decision.
• Test Statistic


  Any function of the observed data whose numerical value dictates whether the null hypothesis is accepted or rejected

  
• 1. test


  • Used when the data are normally distributed, the population standard deviation (σ) is known, and the sample size is greater than or equal to 30 (n ≥ 30)
• 1. test


  • Used when the population is normal / approximately normal, the population standard deviation (σ) is unknown, and the sample size is less than 30 (n < 30)
• Central Limit Theorem (CLT)


  States that if you have a population with mean and standard deviation and take sufficiently large random samples from the population, then the distribution of the sample means will be approximately normally distributed
• Test Statistic Selection


  • T-test: population standard (σ) is unknown, n < 30

  1. test: population standard (σ) is known, n ≥ 30

  Central Limit Theorem (CLT): population standard (σ) is unknown, n ≥ 30

  
• Example 1: Hypothesis testing for employee age

  • Ho: The mean age of the employees in a certain company is 38. Ho: μ = 38

  Ha: The mean age of the employees is not equal to 38. Ha: μ ≠ 38

  α = 5% or 0.05

  Test Statistic: Z-test (population standard deviation is known, n = 30)
• One-tailed test


  A test hypothesis where the alternative hypothesis is one-sided

  
• Two-tailed test


  A test hypothesis where the alternative hypothesis is two-sided
• Rejection region (critical region)


  The set of all values of the test statistic that causes us to reject the null hypothesis

  
• Non-rejection region (acceptance region)


  The set of all values of the test statistic that causes us to fail to reject the null hypothesis

  
• Critical value


  A point (boundary) on the test distribution that is compared to the test statistic to determine if the null hypothesis would be rejected

  
• To get the critical values for the z-test use z-score table.
• To get the critical values for the t-test use t-table.
• If the alternative hypothesis requires the two-tailed test (≠), the alpha is divided by 2 later in determining the critical region.

  
• 1. table


  Table used to determine critical values for t-test
• Probability distributions


  Statistical distributions that describe the probability of different outcomes in a sample
• Formulate the Decision Rule


  Determine the critical region based on the alternative hypothesis and significance level
• Significance levels and corresponding critical values


  • α = 0.1%, cv = ±3.291
  • α = 1%, cv = ±2.576
  • α = 5%, cv = ±1.960
  • α = 10%, cv = ±1.645
• If the alternative hypothesis requires the two-tailed test (≠), the alpha is divided by 2 later in determining the critical region

  
• Null hypothesis (Ho)


  The claim or assumption being tested

  
• Alternative hypothesis (Ha)


  The claim that contradicts the null hypothesis