Hypothesis Testing: Independent and Paired Samples

Hypothesis Testing: Two Samples

This lecture, part of BNAN 562, focuses on hypothesis testing involving two samples, specifically the independent samples t-test and the paired t-test.

Independent Samples t-test

Introduction and Purpose

• The independent samples t-test is used to determine if there is a significant difference between the means of two unrelated (independent) groups.
• Example Question: Are the first-year Northlake MBA GPAs for US students different than for foreign students?

Step 1: Formulate the Null and Alternative Hypotheses

• Null Hypothesis ($H_0$): Assumes no difference between the population means.
  • $H<em>0: \mu</em>{US} - \mu_{foreign} = 0$ (Mean MBA GPAs for US students are not different than foreign students).
• Alternative Hypothesis ($H_a$): Claims there is a difference between the population means.
  • $H<em>a: \mu</em>{US} - \mu_{foreign} \neq 0$ (Mean MBA GPAs for US students are different than foreign students).
• This is a two-tailed test because we are not hypothesizing a specific direction (i.e., not saying US GPAs are higher or lower, just different).
• If a significant difference is found, one would then look at descriptive statistics (means) to determine the direction.

Step 2: Select the Significance Level

• The significance level ($\alpha$) is chosen by comparing the costs of Type I and Type II errors.
• In this example, the traditional alpha level is set at $\alpha = .05$ (5 percent).

Sampling Distribution of Mean Differences

• Every significant test has a sampling distribution in its background.
• For the independent samples t-test, this is the sampling distribution of the differences between the means.
• This distribution shows the probability of observing a difference between two sample means of a certain size, assuming the null hypothesis is true (i.e., there is no real difference between the population means, and any observed difference is due to sampling error).
• Under $H_0$, the expected difference is zero, meaning observed differences close to zero are more likely, and differences further from zero are less likely.
• The $\alpha = .05$ significance level defines two rejection regions (for a two-tailed test), one on each side of the distribution, totaling 5% of the area. These typically start around $\pm 1.97$ standard deviations from the mean (based on the t-table for specific degrees of freedom).
• If the observed difference falls within the middle area, we retain $H<em>0$. If it falls into one of the tails (rejection regions), we reject $H</em>0$ in favor of $H_a$.

Step 3: Select the Statistic and Calculate Its Value

• To compare two unrelated sample means, an Independent Samples t-test is used (often referred to as "Equal Variances" in Excel or "Independent Samples Test" in SPSS).
• The test requires an assumption that the variances of the two populations (US and foreign students) are equal. While a technicality, the t-test is generally robust to violations of this assumption if sample sizes are large and not dramatically different.
• The t-statistic is calculated using the formula: $t = \frac{(\bar{x}<em>1 - \bar{x}</em>2) - (\mu<em>1 - \mu</em>2)}{\sqrt{s<em>p^2 (1/n</em>1 + 1/n_2)}}$ Where:
  • $(\bar{x}<em>1 - \bar{x}</em>2)$ is the observed difference between sample means.
  • $(\mu<em>1 - \mu</em>2)$ is the hypothesized difference under $H_0$ (which is $0$).
  • $s_p^2$ is the pooled sample variance.
  • $n<em>1$ and $n</em>2$ are the sample sizes.
• This process standardizes the observed difference into standard deviation units of the sampling distribution, similar to what's done for one-sample tests.
• Example Calculation: For observed means $\bar{x}<em>1 = 3.18$ (US) and $\bar{x}</em>2 = 3.14$ (Foreign), a pooled variance and sample sizes ($n<em>1 = 172$, $n</em>2 = 30$) yield:
  $t = \frac{(3.18 - 3.14) - 0}{\sqrt{0.103 (1/172 + 1/30)}} = \frac{0.04}{0.063} = 0.64$
• Degrees of Freedom (df): Calculated as $df = n<em>1 + n</em>2 - 2$. For this example, $df = 172 + 30 - 2 = 200$. Two degrees of freedom are lost because two parameters (the average of each sample) are estimated.
• Statistical software (e.g., Excel, SPSS) performs these calculations automatically.

Step 4: Identify the Critical Value for the Test Statistic and State the Decision

• There are two methods to make a decision:
  • Method 1: Using p-value vs. $\alpha$ (Recommended for ease of use with software output)
    • If the two-tailed p-value is less than $\alpha = .05$, reject the null hypothesis. Conclude there are significant differences between the mean MBA GPAs.
    • If the two-tailed p-value is greater than or equal to $\alpha = .05$, retain the null hypothesis. Conclude there is no sufficient evidence of significant differences.
    • If $H<em>0$ is rejected, compare sample means ($\bar{x}</em>{US}$ vs. $\bar{x}_{foreign}$) to determine the direction of the difference.
  • Method 2: Using t-statistic vs. t-critical values
    • Identify the critical t-values ($t<em>{crit}$) from a t-table for the given degrees of freedom and alpha level. For $df = 200$ and $\alpha = .05$ (two-tailed), $t</em>{crit} = \pm 1.97$.
    • If the calculated t-statistic ($|t|$) is greater than $|t_{crit}|$ (i.e., t > 1.97 or t < -1.97 ), reject the null hypothesis.
    • If $|t|$ is less than or equal to $|t_{crit}|$ (i.e., $-1.97 \le t \le 1.97$), retain the null hypothesis.

Step 5: Reaching a Conclusion

• From Excel Output:
  • Calculated t-statistic: $t_{200} = 0.643194$.
  • Two-tailed p-value: $p = 0.520835$.
  • Critical t-value (two-tailed, $\alpha = .05$): $t_{crit} = 1.971896$.
• Using Method 1 (p-value): $p = 0.52$ is greater than $\alpha = .05$. Therefore, retain the null hypothesis.
• Using Method 2 (t-statistic): $t = 0.64$ is between $-1.97$ and $1.97$. Therefore, retain the null hypothesis.
• Conclusion: There is no evidence of a statistically significant difference in the mean MBA GPAs between US and foreign students.

Step 6: Making a Business-Related Decision

• Since no average differences were found between US and foreign students' first-year MBA GPAs, a business decision could be to pool all applicants regardless of citizenship and not use citizenship status as a factor in the selection process.

Notes on "Independent" t-tests

• Nomenclature: