Statistical Inferences Based on Two Samples

Chapter 11: Statistical Inferences Based on Two Samples

Chapter Outline

• 11.1 Comparing Two Population Means by Using Independent Samples
• 11.2 Paired Difference Experiments
• 11.3 Comparing Two Population Proportions by Using Large, Independent Samples
• 11.4 The F Distribution
• 11.5 Comparing Two Population Variances by Using Independent Samples

11.1 Comparing Two Population Means by Using Independent Samples

• Assumptions:
  • A random sample is taken from each of two different populations.
  • The populations are independent of each other.
  • The random samples are also independent of each other.
• The sampling distribution of the difference in sample means is normally distributed.
  • Learning Objective LO11-1: Compare two population means when the samples are independent.

Sampling Distribution of the Difference of Two Sample Means

1. Hypotheses: Populations Characteristics
   • Population 1: Mean = $\mu<em>1$, Variance = $\sigma</em>1^2$
   • Sample drawn: size = n1, Mean = $\bar{x}<em>1$, Variance = $s</em>1^2$
   • Population 2: Mean = $\mu<em>2$, Variance = $\sigma</em>2^2$
   • Sample drawn: size = n2, Mean = $\bar{x}<em>2$, Variance = $s</em>2^2$
2. Properties of the Sampling Distribution:
   • It is normal if each population sampled is normal.
   • It is approximately normal if sample sizes n1 and n2 are large.
   • Mean of the Distribution: $\mu<em>{\bar{x}</em>1 - \bar{x}<em>2} = \mu</em>1 - \mu_2$
   • Standard Deviation of the Distribution: $\sigma<em>{\bar{x}</em>1 - \bar{x}<em>2} = \sqrt{\frac{\sigma</em>1^2}{n<em>1} + \frac{\sigma</em>2^2}{n_2}}$

Pooled Estimate of $\sigma^2$

• Assumption: $\sigma<em>1^2 = \sigma</em>2^2 = \sigma^2$
• Pooled variance estimate:
  $s<em>p^2 = \frac{(n</em>1-1)s<em>1^2 + (n</em>2-1)s<em>2^2}{n</em>1 + n_2 - 2}$
• Estimate of population standard deviation of sampling distribution:
  $\sigma<em>{\bar{x}</em>1 - \bar{x}<em>2} = \sqrt{s</em>p^2(\frac{1}{n<em>1} + \frac{1}{n</em>2})}$

Confidence Interval for the Difference in Means (Equal Variances)

1. Selecting Samples:
   • Independent random samples from two normal populations with equal variances.
2. Confidence Interval Formula:
   $\left(\bar{x}<em>1 - \bar{x}</em>2 \right) \pm t<em>{\alpha/2} s</em>p \sqrt{\frac{1}{n<em>1} + \frac{1}{n</em>2}}$
   where $df = n<em>1 + n</em>2 - 2$

Test Statistic for Differences in Means (Equal Variances)

• Formula for Test Statistic:
  $t = \frac{(\bar{x}<em>1 - \bar{x}</em>2) - D<em>0}{s</em>p \sqrt{\frac{1}{n<em>1} + \frac{1}{n</em>2}}}$
  where $D<em>0 = \mu</em>1 - \mu_2$ (claimed difference).
• The sampling distribution of this statistic follows a t distribution with df = $n<em>1 + n</em>2 - 2$.

Hypothesis Testing for the Difference Between Two Means (Equal Variances)

• Null Hypothesis (H0): $\mu<em>1 - \mu</em>2 = D_0$
• Assumptions for testing:
  1. Independent samples.
  2. Equal variances or large sample sizes.

11.2 Paired Difference Experiments

• Paired difference experiments involve two different processes or methods.
• A random sample of units is used, and those same units are utilized to compare both processes.
  • This method eliminates individual differences between units.
• Mean of paired differences: $\mu<em>d = \mu</em>1 - \mu_2$
  • $\bar{d}$ is the mean of measured differences from paired samples.

Confidence Interval for Paired Differences

• If differences in population of paired samples are normally distributed, the confidence interval for mean of differences is:
  $[\bar{d} \pm t<em>{\alpha/2} \frac{s</em>d}{\sqrt{n}}]$
  where $d$ is the paired difference, and $n$ is the number of pairs.

Hypothesis Testing about Mean of Paired Differences

• Test Statistic Formula:
  $t = \frac{\bar{d} - D<em>0}{s</em>d / \sqrt{n}}$
• Null Hypothesis: $H<em>0: \mu</em>d = D<em>0$; often set as $D</em>0 = 0$.
• The sampling distribution follows a t distribution with $n - 1$ degrees of freedom.

11.3 Comparing Two Population Proportions by Using Large, Independent Samples

• Let $p<em>1$ be the sample proportion of units in a category from population 1 and $p</em>2$ for population 2.
  • Conditions required:
  • $n<em>1 p</em>1 \geq 5$, $n<em>1 (1 - p</em>1) \geq 5$, $n<em>2 p</em>2 \geq 5$, $n<em>2 (1 - p</em>2) \geq 5$
• Mean of the Difference: $\mu<em>{p</em>1 - p<em>2} = p</em>1 - p_2$
• Standard Deviation of the Difference:
  $\sigma<em>{p</em>1 - p<em>2} = \sqrt{\frac{p</em>1(1 - p<em>1)}{n</em>1} + \frac{p<em>2(1 - p</em>2)}{n_2}}$

Confidence Interval for Difference in Proportions

• If independent samples, the confidence interval is given by:
  $[ (p<em>1 - p</em>2) \pm z<em>{\alpha/2} \sigma</em>{p<em>1 - p</em>2}]$

Test Statistic for the Difference of Two Population Proportions

• Test Statistic Formula: z = \frac{(p1 - p2) - D0}{\sqrt{\frac{p(1 - p)(\frac{1}{n1} + \frac{1}{n_2})}}}
  • $D_0$ is the claimed difference in proportions, often set as zero.
  • The distribution is standard normal if the sample sizes are large enough.

11.4 The F Distribution

• The F distribution is characterized by two parameters: degrees of freedom for the numerator (df1) and denominator (df2).
• It is skewed to the right, indicating that it has a non-symmetrical distribution.
• The F point, $F_a$, corresponds to a right-tail area equal to $α$, depending on df1 and df2.

Testing Equality of Population Variances

• Hypothesis: For comparing variances, null hypothesis $H<em>0: \sigma</em>1^2 = \sigma_2^2$.
• The test statistic is defined as: $F = \frac{s<em>1^2}{s</em>2^2}$
  • This will be significantly greater than one under the null.

11.5 Comparing Two Population Variances by Using Independent Samples

• The null hypothesis for variances: $H<em>0: \sigma</em>1^2 = \sigma_2^2$.
• The alternative hypothesis suggests one population has less variability than the other.
• The sampling distribution for $\frac{s<em>1^2}{s</em>2^2}$ follows an F distribution.

These notes cover several fundamental aspects of statistical inference based on two samples, particularly concerning means, proportions, and variances. The details provided are structured for comprehensive understanding and quick reference for study purposes.