In a batch chemical process, two catalysts arc being...

In a batch chemical process, two catalysts arc being compared for their effect on the output of the process reaction. A sample of $\lfloor 2$ batches was prepared using catalyst 1 and a sample of 10 batches was obtained using catalyst $2 .$ The 12 batches for which catalyst 1 was used gave an average yield of 85 with a sample standard deviation of $4,$ and the second sample gave an average of 81 and a sample standard deviation of $5 .$ Find a $90 \%$ confidence interval for the difference between the population means, assuming that the: populations art: approximately normally distributed with equal variances.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. In the context of comparing means, it provides an estimated range for the difference between the two population means with a specified level of confidence, such as 90% in this case.

Two-Sample t-Test

The two-sample t-test is a statistical method used to compare the means of two independent groups. It is particularly useful when sample sizes are small and when the populations are assumed to be normally distributed, enabling analysts to determine if observed differences between sample means are statistically significant.

Equal Variance Assumption

The equal variance assumption posits that the two populations being compared have the same variance. When this assumption holds true, a pooled variance estimate can be used, which combines data from both samples to provide a more reliable estimate of the common variance, improving the precision of the confidence interval or hypothesis test.

Pooled Variance

Pooled variance is a weighted average of the sample variances from two independent samples. It combines the information from both groups under the assumption that their variances are equal, and is used in the calculation of the standard error when constructing the confidence interval or performing a t-test.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in the analysis without breaking any constraints. In the context of a two-sample t-test with equal variances, the degrees of freedom are typically calculated as the sum of the sample sizes of the two groups minus 2. This value is essential for determining the appropriate critical value from the t-distribution when constructing the confidence interval.

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