Two catalysts may be used in a batch chemical process. Twelve batches were prepared using catalyst 1, resulting in an average yield of 85 and a sample standard deviation of 3. Fifteen batches were prepared using catalyst 2, and they resulted in an average yield of 91 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 the claim that catalyst 2 produces 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). (e. g. 98.76). mu_1 - mu_2 lessthanorequalto the tolerance is +/-2%

a)  

So with the p value obtained and using the significance level given we have so we can conclude that we have enough evidence to reject the null hypothesis.

b)

Step-by-step explanation:

Notation and hypothesis

When we have two independent samples from two normal distributions with equal variances we are assuming that  

And the statistic is given by this formula:  

Where t follows a t distribution with degrees of freedom and the pooled variance is given by this formula:  

This last one is an unbiased estimator of the common variance  

Part a

The system of hypothesis on this case are:  

Null hypothesis:  

Alternative hypothesis:  

Or equivalently:  

Null hypothesis:  

Alternative hypothesis:  

Our notation on this case :  

represent the sample size for group 1  

represent the sample size for group 2  

represent the sample mean for the group 1  

represent the sample mean for the group 2  

represent the sample standard deviation for group 1  

represent the sample standard deviation for group 2  

First we can begin finding the pooled variance:  

And the deviation would be just the square root of the variance:  

Calculate the statistic

And now we can calculate the statistic:  

Now we can calculate the degrees of freedom given by:  

Calculate the p value

And now we can calculate the p value using the altenative hypothesis:  

Conclusion

So with the p value obtained and using the significance level given we have so we can conclude that we have enough evidence to reject the null hypothesis.

Part b

For this case the confidence interval is given by:

For the 99% of confidence we have and and the critical value with 25 degrees of freedom on the t distribution is

And replacing we got:

answer is 8 r5

step-by-step explanation: