intro to probability for business unit 9 study guides

Key Concepts

• Two-sample hypothesis tests compare parameters (means, proportions, or variances) between two independent populations
• Null hypothesis ($H_0$) assumes no significant difference between the two population parameters
• Alternative hypothesis ($H_a$) suggests a significant difference exists between the two population parameters
• Type I error (α) occurs when rejecting a true null hypothesis (false positive)
• Type II error (β) happens when failing to reject a false null hypothesis (false negative)
  • Decreasing α increases β, and vice versa, creating a trade-off between the two error types
• Statistical significance is determined by comparing the p-value to the chosen significance level (α)
  • If p-value ≤ α, reject $H_0$ and conclude a significant difference exists
  • If p-value > α, fail to reject $H_0$ and conclude insufficient evidence of a significant difference
• Power of a test (1 - β) represents the probability of correctly rejecting a false null hypothesis

Types of Two-Sample Tests

• Two-sample t-test compares means between two independent populations with normal distributions
  • Independent samples t-test assumes equal variances between the two populations
  • Welch's t-test is used when population variances are unequal or unknown
• Two-sample z-test for proportions compares proportions between two independent populations
• F-test compares variances between two independent populations with normal distributions
• Mann-Whitney U test (Wilcoxon rank-sum test) is a non-parametric alternative to the two-sample t-test
  • Used when data is ordinal or when normality assumption is violated
• Chi-square test for homogeneity compares proportions between two or more independent populations
• Paired t-test compares means between two dependent samples (before and after measurements)

Assumptions and Conditions

• Independence assumption requires that the two samples are randomly selected and independent of each other
  • Violation of independence can lead to biased results and invalid conclusions
• Normality assumption states that the populations from which the samples are drawn follow a normal distribution
  • For large sample sizes (n > 30), the Central Limit Theorem allows for approximation of normality
  • Non-parametric tests (Mann-Whitney U test) can be used when normality is violated
• Equal variances assumption (for independent samples t-test) assumes that the two populations have equal variances
  • Levene's test or F-test can be used to assess equality of variances
  • Welch's t-test is an alternative when variances are unequal or unknown
• Random sampling ensures that the samples are representative of their respective populations
• Sample size requirements vary depending on the test and desired power
  • Larger sample sizes generally increase the power of the test and reduce the impact of violations of assumptions

Test Statistics and Formulas

• Independent samples t-test statistic: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
  • $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $s_1^2$ and $s_2^2$ are the sample variances, and $n_1$ and $n_2$ are the sample sizes
• Welch's t-test statistic: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$, with adjusted degrees of freedom
• Two-sample z-test for proportions statistic: $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$
  • $\hat{p}_1$ and $\hat{p}_2$ are the sample proportions, and $\hat{p}$ is the pooled proportion
• F-test statistic: $F = \frac{s_1^2}{s_2^2}$, where $s_1^2$ and $s_2^2$ are the sample variances
• Mann-Whitney U test statistic: $U = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1$ or $U = n_1n_2 + \frac{n_2(n_2+1)}{2} - R_2$
  • $n_1$ and $n_2$ are the sample sizes, and $R_1$ and $R_2$ are the ranks of the observations

Interpreting Results

• P-value represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true
  • A small p-value (≤ α) suggests strong evidence against the null hypothesis, leading to its rejection
  • A large p-value (> α) indicates insufficient evidence to reject the null hypothesis
• Confidence intervals provide a range of plausible values for the difference between the two population parameters
  • A 95% confidence interval means that if the sampling process were repeated many times, 95% of the intervals would contain the true difference
  • If the confidence interval includes zero, it suggests no significant difference between the two populations
• Effect size measures the magnitude of the difference between the two populations
  • Cohen's d is commonly used for comparing means, with values of 0.2, 0.5, and 0.8 indicating small, medium, and large effects, respectively
  • Odds ratio or relative risk is used for comparing proportions, with values farther from 1 indicating a stronger association
• Practical significance considers the real-world implications of the results, beyond statistical significance
  • A statistically significant difference may not always be practically meaningful, depending on the context and the magnitude of the effect

Common Pitfalls

• Failing to check assumptions and conditions before conducting the test
  • Violations of assumptions can lead to incorrect test results and invalid conclusions
• Using the wrong test for the given data or research question
  • Choosing an inappropriate test can result in misleading or inaccurate findings
• Misinterpreting the p-value as the probability of the null hypothesis being true
  • The p-value is the probability of observing the data, assuming the null hypothesis is true, not the other way around
• Confusing statistical significance with practical significance
  • A statistically significant result may not always be practically meaningful or actionable
• Failing to consider the power of the test and the potential for Type II errors
  • A non-significant result may be due to insufficient power, rather than a true lack of difference between the populations
• Multiple testing issues, such as inflated Type I error rates when conducting multiple comparisons
  • Bonferroni correction or other methods can be used to adjust for multiple testing

Real-World Applications

• A/B testing in marketing compares the effectiveness of two different versions of a website or advertisement
  • Two-sample t-test or z-test can be used to compare conversion rates or other metrics between the two versions
• Clinical trials in medical research compare the efficacy of a new treatment to a standard treatment or placebo
  • Two-sample t-test or Mann-Whitney U test can be used to compare patient outcomes between the two groups
• Quality control in manufacturing compares the defect rates or product measurements between two production lines or suppliers
  • Two-sample z-test for proportions or F-test for variances can be used to identify significant differences
• Customer satisfaction surveys compare ratings or feedback between two different customer segments or time periods
  • Two-sample t-test or chi-square test can be used to analyze differences in satisfaction levels or response patterns
• Educational research compares test scores or learning outcomes between two different teaching methods or curricula
  • Two-sample t-test or paired t-test can be used to assess the effectiveness of the interventions

Practice Problems

1. A company wants to compare the mean sales between two store locations. A random sample of 50 sales transactions from each store yielded the following results:

   • Store A: $\bar{x}_A = $75$, $s_A = $20$
   • Store B: $\bar{x}_B = $80$, $s_B = $25$ Conduct a two-sample t-test at the 5% significance level to determine if there is a significant difference in mean sales between the two stores.

2. A medical researcher wants to compare the effectiveness of two different treatments for a specific condition. In a randomized controlled trial, 100 patients were assigned to each treatment group. The success rates were:

   • Treatment A: 75 out of 100 patients improved
   • Treatment B: 65 out of 100 patients improved Use a two-sample z-test for proportions at the 1% significance level to determine if there is a significant difference in success rates between the two treatments.

3. A manufacturing company wants to compare the variability in product weights between two production lines. Random samples of 30 products from each line were measured, with the following results:

   • Line 1: $s_1^2 = 4.2$ ounces²
   • Line 2: $s_2^2 = 6.5$ ounces² Perform an F-test at the 10% significance level to determine if there is a significant difference in variances between the two production lines.

4. A psychologist wants to compare the effectiveness of two different therapy techniques for reducing anxiety. In a matched-pairs design, 20 patients were assessed before and after receiving each therapy, with the following results:

   • Therapy A: $\bar{d}A = 10$ points, $s{d_A} = 6$ points
   • Therapy B: $\bar{d}B = 7$ points, $s{d_B} = 5$ points Conduct a paired t-test at the 5% significance level to determine if there is a significant difference in anxiety reduction between the two therapies.

5. A market researcher wants to compare customer satisfaction ratings between two competing brands. Independent random samples of customers were surveyed, with the following results:

   • Brand A: $n_A = 200$, median rating = 4 out of 5
   • Brand B: $n_B = 180$, median rating = 3 out of 5 Use the Mann-Whitney U test at the 1% significance level to determine if there is a significant difference in customer satisfaction between the two brands.