Two catalysts may be used in a batch chemical process....

Two catalysts may be used in a batch chemical process. Twelve batches were prepared using catalyst $1,$ resulting in an average yield of 86 and a sample standard deviation of $3 .$ Fifteen batches were prepared using catalyst $2,$ and they resulted in an average yield of 89 with a standard deviation of 2 . Assume that yield measurements are approximately normally distributed with the same standard deviation. (a) Is there evidence to support a claim that catalyst 2 produces a higher mean yield than catalyst $1 ?$ Use $\alpha=0.01$. (b) Find a $99 \%$ confidence interval on the difference in mean yields that can be used to test the claim in part (a).

Two catalysts may be used in a batch chemical process. Twelve batches were prepared using catalyst $1,$ resulting in an average yield of 86 and a sample standard deviation of $3 .$ Fifteen batches were prepared using catalyst $2,$ and they resulted in an average yield of 89 with a standard deviation of 2 . Assume that yield measurements are approximately normally distributed with the same standard deviation.
(a) Is there evidence to support a claim that catalyst 2 produces a higher mean yield than catalyst $1 ?$ Use $\alpha=0.01$.
(b) Find a $99 \%$ confidence interval on the difference in mean yields that can be used to test the claim in part (a).

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In the context of comparing two means, it often involves formulating a null hypothesis that states there is no difference (or a specific direction of no difference) between the groups, and an alternative hypothesis that indicates a difference (or a directional difference) exists. This method helps determine if any observed differences in outcomes can be attributed to chance or reflect a true effect.

Two-Sample t-Test

The two-sample t-test is a statistical procedure used to determine whether there is a significant difference between the means of two independent groups. It is particularly useful when sample sizes are small and the population standard deviations are unknown, but assumed to be equal. The test statistic follows a t-distribution when the data are approximately normally distributed and the assumption of equal variances holds.

Pooled Variance

When assuming that two independent samples have equal population variances, the pooled variance is computed as a weighted average of the individual sample variances. This approach increases the reliability of the variance estimate by combining information from both samples, and it is used in calculating the test statistic for the two-sample t-test.

Degrees of Freedom

Degrees of freedom in the context of the two-sample t-test refer to the number of independent pieces of information available to estimate the population variance. When using the pooled variance, the degrees of freedom are typically derived from the total number of observations across both groups minus the number of groups. This value is essential to determine the critical values from the t-distribution for hypothesis testing and confidence interval estimation.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used in hypothesis testing to determine whether to reject the null hypothesis. It represents the probability of making a Type I error, which is the error of rejecting a true null hypothesis. By setting a significance level, researchers control the risk of falsely detecting an effect where none exists.

Confidence Interval for the Difference in Means

A confidence interval for the difference in means provides a range of values within which the true difference is expected to lie with a specified level of confidence (such as 99% in this case). This interval not only offers a measure of the precision of the estimated difference but also serves as an inferential tool; if the interval does not include zero, it suggests that there is a statistically significant difference between the group means.

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