A pharmaceutical company is interested in testing the effect of humidity on the weights of pills sold in a new aluminum package. Let X and Y denote the weight of a pill in the old aluminum package and in the new aluminum package (respectively) after the packaged pill has spent one week in chamber kept at 30 °C and 100 % humidity. Define a null and alternates hypothesis to test whether the old aluminum package pills weigh less than the new aluminum package pills following the humidity-chamber treatment. The following random samples of X yielded the following weights in milligrams: 373.2 376.7 381.6 382.1 388.7 384.0
397.9 389.8 385.1 371.3 383.5
and the following random sample of Y yielded the following weights in milligrams:
395.5 384.8 383.5 386.5 394.8
391.6 397.7 384.0 391.7 398.8
Define a critical region with a significance level of alpha = 0.05 and calculate the value of the test statistic. Accept or reject the null hypothesis and then state a scientific conclusion about the packaged-pill humidity experiment.

The null hypothesis is  

The alternative hypothesis is  

The test statistics is  

Reject the null hypothesis

There is  sufficient evidence to conclude that the weight of the old aluminium package pills is less than the new aluminium pills.

Step-by-step explanation:

From the question we are told that

The data for X is  373.2 , 376.7  , 381.6 , 382.1  , 388.7 ,  384.0 ,  397.9 , 389.8 ,385.1 , 371.3 , 383.5

The data for Y is 395.5 ,  384.8 , 383.5  , 386.5  ,394.8

, 391.6  , 397.7 , 384.0 , 391.7 , 398.8

Generally the sample mean for X is mathematically represented as

=>      

=>      

Generally the sample mean for Y is mathematically represented as

=>      

=>      

Generally the standard deviation for  X is mathematically represented as

=>  

=>  

Generally the standard deviation for  Y is mathematically represented as

=>  

=>  

The null hypothesis is  

The alternative hypothesis is  

Generally the test statistics is mathematically represented as

=>  

=>  

Generally the level of significance is

Generally the degree of freedom is mathematically represented as

=>  

=>  

Generally from the student t- distribution table the critical value of   at a df = 19 is      

Now comparing the critical value  obtained with test statistic calculated we see that the critical value is the region of the calculated test statistic( i.e  between  -2.65 and  2.65) hence we reject the null hypothesis

Therefore there sufficient evidence to conclude that the old aluminium package pills weight is less than the new aluminium pills

the correct answer is 828.07