Use Table 5 in Appendix B to find the critical value(s) for...

Use Table 5 in Appendix $B$ to find the critical value(s) for the alternative hypothesis, level of significance $\alpha,$ and sample sizes $n_{1}$ and $n_{2}$. Assume that the samples are random and independent, the populations are normally distributed, and that the population variances are (a) equal and (b) not equal. $$H_{a}: \mu_{1} \neq \mu_{2}, \alpha=0.10, n_{1}=11, n_{2}=14$$

Use Table 5 in Appendix $B$ to find the critical value(s) for the alternative hypothesis, level of significance $\alpha,$ and sample sizes $n_{1}$ and $n_{2}$. Assume that the samples are random and independent, the populations are normally distributed, and that the population variances are (a) equal and (b) not equal.
$$H_{a}: \mu_{1} \neq \mu_{2}, \alpha=0.10, n_{1}=11, n_{2}=14$$

Key Concepts

Two-Sample t-Test

This is a statistical procedure used to determine if there is a significant difference between the means of two independent groups. It is applicable when the populations are normally distributed and is the primary framework for comparing the means under different assumptions about the variances.

Hypothesis Testing Framework

This involves setting up a null hypothesis (that there is no difference between the population means) and an alternative hypothesis (here, that the means are not equal). Testing is conducted by comparing a test statistic to critical values to decide whether to reject the null hypothesis.

Level of Significance (?)

The level of significance, denoted by ?, is the threshold for making a decision in hypothesis testing. It represents the probability of committing a Type I error, that is, rejecting the null hypothesis when it is actually true. In this context, ? is set to 0.10.

Equal Variance Assumption (Pooled Variance t-Test)

When the assumption is made that the two populations have equal variances, the test uses a pooled estimate of variance and a combined degrees of freedom (n1 + n2 - 2) to determine the critical t-value. This method increases power when the equal variance assumption holds.

Unequal Variance Assumption (Welch's t-Test)

If the variances are assumed to be unequal, a modified t-test, often called Welch's t-test, is used. This approach does not pool the variance and employs a different formula for degrees of freedom, which often results in a non-integer value and a slightly different critical value.

Critical Values

Critical values are the cutoff points determined from statistical tables (such as Table 5 in Appendix B) using the relevant degrees of freedom and the specified significance level. For a two-tailed test, these values define the rejection regions for the test statistic in the tail areas of the t-distribution.

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