9.2 Two Means (Independent)

Introduction

This section covers the comparison of two population means using hypothesis testing and confidence intervals, specifically when the samples are independent and the population standard deviations are unknown. These methods are fundamental in inferential statistics for determining whether two groups differ significantly in their means.

Independent Samples, σ1 and σ2 Unknown

When comparing two means, it is common that the population standard deviations are unknown. In such cases, we use sample data to estimate these values and apply statistical tests to draw conclusions about the populations.

Definition

• Independent samples: Samples from different populations that have data values not related to one another.

• Dependent (or matched pairs) samples: Samples from different populations that have data values related or matched to one another.

Hint: If samples have different sample sizes with no missing data, the samples are independent. If the sample sizes are the same, they may be dependent.

Example 1: Dependent Samples

The following table lists Oscar winners by age for best actress and actor by year:

Since ages are listed per year, they are related by the year and thus are dependent samples.

Example 2: Independent Samples

The following table lists the volumes of soda cans for ten different cans of Coke and Pepsi:

These samples are not related to one another and are therefore independent.

Inferences about Two Means (Independent)

Objectives

• Hypothesis Test about a claim concerning two means.

• Construct a confidence interval estimate about the difference between two means.

Requirements

• Samples are from simple random samples.

• Samples are independent.

• Sample sizes and , or both samples come from normally distributed populations.

• Population standard deviations and are unknown.

Note: The requirement for normality is robust; if sample sizes are nearly 30 or distributions are nearly normal, the approximation works well.

Notation

• : Population mean for group 1

• : Sample mean for group 1

• : Sample standard deviation for group 1

• : Sample size for group 1

• Similar notation applies for group 2: , , ,

• : Degrees of freedom, smaller of and

Test Statistic

The test statistic for comparing two independent means is:

P-value and Critical Values

• P-value: Found using calculator or software.

• Critical Values: Computed with calculator or software.

Confidence Interval Estimate of

The confidence interval estimate of the difference is:

Where is the margin of error:

Confidence interval limits are rounded to three significant digits.

Hypothesis Test

When testing a claim about two means, the null hypothesis is always:

The alternative hypothesis can be:

• (two-tailed)

• (right-tailed)

• (left-tailed)

Example 3: Hypothesis Test and Confidence Interval

Researchers tested whether color affects creativity. Subjects were divided into two groups: one with a red background and one with a blue background. Creativity scores were measured as follows:

• Test the claim that blue enhances performance on a creative task at a 0.01 significance level.

• Test the same claim with a confidence interval.

Let:

• : Mean creativity score with red background

• : Mean creativity score with blue background

• , ,

• , ,

Since both samples are greater than 30, requirements are met.

Null and alternative hypotheses:

Test statistic and p-value are calculated (using calculator/software):

Since , we reject the null hypothesis. There is evidence to support that the mean creativity scores with blue backgrounds are higher than those with red backgrounds.

Confidence Interval Estimate

Using a 2-SampTInt, the confidence interval is:

Since the interval does not contain 0 and all numbers are negative, we conclude that , supporting the rejection of the null hypothesis.

Critical Value Method

For t distributions, finding critical values is more involved or requires use of a table. For means, hypothesis testing with p-values, critical values, and confidence intervals will provide equivalent results.

Example 4: Application

Record your pulse rate over 1 minute and post results split into two groups: males and females. Pulse rates are normally distributed. Test the claim that the pulse rate of females is higher than that of males using both p-value and confidence interval methods at the 0.05 significance level.

Summary Table: Comparison of Independent and Dependent Samples

Key Points

• Use independent sample t-tests when comparing means from two unrelated groups.

• Check requirements: random sampling, independence, normality or large sample size, unknown population standard deviations.

• Formulate null and alternative hypotheses based on the research question.

• Calculate test statistic and p-value to make decisions about the hypotheses.

• Construct confidence intervals to estimate the difference between means.

• Interpret results in the context of the problem.

Additional info: For more advanced cases, pooled variance may be used if population variances are assumed equal. For small sample sizes, normality should be checked with graphical or statistical methods.