Statistical Methods Notes

1. Parametric Hypothesis Tests for a Mean

1.1 Learning Goals

• Understanding the parametric hypothesis test for a mean.

• Understanding the concept of effect size.

2. Hypothesis Testing Procedure

2.1 Stating Hypotheses

• In context of testing the population mean $heta$, possible hypothesis statements include:

  • Null Hypothesis: $H_0: heta = heta_0$

  • Alternative Hypotheses:

  • $H_A: heta <br>eq heta_0$ (two-tailed test)

  • H_A: heta < heta_0 (left-tailed test)

  • H_A: heta > heta_0 (right-tailed test)

2.2 Checking Conditions

• Conditions that must be satisfied for valid results:

  1. Independence: The observations must be independent.

  2. Sample Size: The sample size should be large enough (n ≥ 30). This condition may be relaxed if the observations are drawn from a normally distributed population.

2.3 Computing Test Statistic and Finding the p-value

• Test Statistic Formula: $t = rac{\bar{x} - heta_0}{ rac{s}{\sqrt{n}}}$ Where:

  • $\bar{x}$ is the sample mean,

  • $heta_0$ is the hypothesized population mean,

  • $s$ is the sample standard deviation,

  • $n$ is the sample size.

• Finding p-value:

  • The p-value represents the area beyond the test statistic $t$ in the direction specified by the alternative hypothesis in a t-distribution with $n - 1$ degrees of freedom.

• The area in a t-distribution is computed using the function tcdf() in statistical software.

2.4 Evaluating Results

• Criteria for interpreting the p-value:

  • p > 0.10: Little evidence against $H_0$

  • 0.05 < p \leq 0.10: Some evidence against $H_0$

  • 0.01 < p \leq 0.05: Strong evidence against $H_0$

  • 0.001 < p \leq 0.01: Very strong evidence against $H_0$

  • $p \leq 0.001$: Extremely strong evidence against $H_0$

3. Practical Example: Customer Payments

3.1 Research Context

• Scenario: Adam, an accountant, is concerned about increasing days customers take to make payments.

• Historical Mean: Previous evaluations indicate a mean outstanding time of approximately 50 days.

• Current Sample: A random sample of 35 invoices reveals $\bar{x} = 55$ days, with a standard deviation of $s = 6.9$ days.

3.2 Research Question

1. State the null and alternative hypotheses needed to address Adam's concern.

   • Null Hypothesis ($H_0$): $heta = 50$

   • Alternative Hypothesis ($H_A$): heta > 50

3.3 Checking Conditions for the Hypothesis Test

• a. Independence: Are the observations independent?

• b. Normal Distribution Assumption: Can we assume the population is normally distributed? Does it matter in this situation?

3.4 Calculating the Test Statistic and p-value

• Test Statistic Calculation:
  $t = rac{\bar{x} - 50}{ rac{6.9}{\sqrt{35}}} = -4.2870$

• p-value Calculation:

  • Using statistical software:
    $p = ext{tcdf}(4.287, 10^{10}, 34) = 0.00007$

3.5 Evaluating the p-value

• Result:

  • $p ext{-value} = 0.00007$ which indicates extremely strong evidence against the null hypothesis concerning the mean outstanding time.

4. Effect Size

4.1 Definition

• Effect Size measures the magnitude of difference/relationship in a study.

• For hypothesis testing concerning population means:
  $d = rac{ heta - heta_0}{ au}$
  where $au$ is the population standard deviation.

4.2 Estimating Effect Size

• As population trends are often unknown, we can estimate Cohen's d using sample statistics:
  $ilde{d} = rac{\bar{x} - heta_0}{s}$

4.3 Example: From Adam's Test

• Sample size: n = 35;

• Sample mean: $\bar{x} = 55$;

• Sample standard deviation: $s = 6.9$;

• Therefore, the estimated effect size is:
  $ilde{d} = rac{55 - 50}{6.9}$
  (Calculation details can be provided.)

4.4 Effect Size Interpretation Table

4.5 Effect Size Caution

• Interpretation of effect size as 'small', 'moderate', or 'large' can be context-driven.

• Example implications:

  • Reading score effect sizes vary significantly with context.

  • An effect size of $| ilde{d}| = 0.03$ may be negligible while $| ilde{d}| = 0.97$ indicates a substantial outcome.

5. Additional Examples

5.1 Example 1

• Null Hypothesis: $H_0: heta = 20$

• Alternative Hypothesis: H_A: heta < 20

• Sample: n = 50 with $\bar{x} = 16$, $s = 5$. Compute and interpret the effect size.

5.2 Example 2

• Null Hypothesis: $H_0: heta = 1.5$

• Alternative Hypothesis: $H_A: heta <br>eq 1.5$

• Sample: n = 10 with $\bar{x} = 1.6$, $s = 3$. Compute and interpret the effect size.