Two different antibody treatments, MVC and PVy each produce an effect that lessens the loss of smell common in Covid-19 patients. Let us call these effects MVC and PVy. MVC and PVy measure smell acuity for peppermint using the TOSI (test of smell identification). We will study the change in smell acuity by measuring the magnitude of peppermint smell gain on the TOSI after the antibody treatments. The true (unknown) mean amounts of peppermint smell gain (on TOSI) is?_x

for the MVC method and? _y

for the PVy method. The treatment method allows us to determine the true standard deviations of smell gain for the MVC and the PVy methods. It is recognized that the true standard deviations are?_x

= 3.01 for the MVC method and?_y

= 2.3 for the PVy method. We took a random sample of m = 71 MVx treated individuals and n=81 PVy treated individuals. The mean amounts of peppermint smell gain werex= 13.33 for MVxindividuals andy= 15.00 for PVy individuals. We are thinking of treating individuals with both full doses of MVx and full doses of PVy. It is known that TOSI measurements are normally distributed. Assuming independence and additivity of effect for the antibody treatments we would like to estimate?_x+?_y

.

a) What is the standard deviation of the distribution ofx?

b) What is the standard deviation of the distribution ofx+y?

c) Create a 92% confidence interval for?_x+?_y

?

d) What is the length of the confidence interval in part c) ?

e) If we let n stay at 81 but vary m, what is the smallest m for which the length of the 90% confidence interval would be 1.10 or less?

f) Copy your R script for the above into the text box here.