Question

A dairy scientist is testing a new feed additive. She chooses 13 cows at random from a large population. She randomly assigns n_old = 8 to the old diet and n_new = 5 to a new diet including the additive. The cows are housed in 13 widely separated pens. After two weeks, she milks each cow and records the milk produced in pounds:

Old Diet: 43, 51, 44, 47, 38, 46, 40, 35 New Diet: 47, 75, 85, 100, 58

Let μnew and μold be the population mean milk productions for the new and old diets, respectively. She wishes to test H0 : μnew − μold = 0 against HA : μnew − μold ̸= 0 using α = 0.05.

(a) Graph the data as you see fit. Why did you choose the graph(s) that you did and what does it (do they) tell you?

(b) Choose a test appropriate for the hypotheses and justify your choice based on your answer to part (a). Then perform the test by computing a p-value, and making a reject or not reject decision. Do this without R and show your work. (Also do it with R, if you wish, to check your work). Finally, state your conclusion in the context of the problem.

Answer 1

(a) The graph is:

I chose this graph to check the variation in milk produced rate. It is clear that the New Diet is way better than the Old Diet.

(b) The hypothesis being tested is:

H0 : μnew − μold = 0 against HA : μnew − μold ̸= 0

The output is:

Group Statistics                  
   Group   N   Mean   Std. Deviation   Std. Error Mean
Diet   1   8   43.0000   5.18239   1.83225
   2   5   73.0000   21.08317   9.42868

Independent Samples Test                                      
       Levene's Test for Equality of Variances       t-test for Equality of Means                      
       F   Sig.   t   df   Sig. (2-tailed)   Mean Difference   Std. Error Difference   95% Confidence Interval of the Difference  
                                   Lower   Upper
Diet   Equal variances assumed   10.552   .008   -3.936   11   .002   -30.00000   7.62144   -46.77468   -13.22532
   Equal variances not assumed           -3.123   4.304   .032   -30.00000   9.60506   -55.94058   -4.05942

Since the p-value (0.002) is less than the significance level, we can reject the null hypothesis.

Therefore, we can conclude that the new diet and the old diet have a significant effect on cows milk production.