Question

***This problem must be done using R so please provide the R code used to find the solution. I have provided the data in Preservative file below the question. I will also give "thumbs-up for correct R code" Thanks in advance.***

Chemical preservatives are often used to extend the shelf life of bakery products such as bread, muffins, buns, etc. There are several types of safe and effective preservatives that are widely used in the baking industry. In a study to test the effectiveness of commonly-used chemical preservatives, two loaves of bread were baked using identical ingredients apart from the preservative used in each loaf. The bread was then sliced and the pieces were inspected for mold. The amount of time (in days) before the pieces started showing visible mold are recorded in the Preservative file emailed to you along with this quiz. (a) [1 point] Construct a 98% confidence interval for ups – USB, the average difference in the amount of time it takes for bread with the preservative potassium sorbate (PS) to grow mold and the amount of time it takes for bread with the preservative sodium benzoate (SB) to grow mold. State your interval in the space below. It is recommended that you use R to compute this interval; if you do, you can just write down the interval you obtain. If you wish to calculate it by hand, please show all of your work. Round your interval bounds to four decimal places.

Here is the rest of the data from the Preservative file to solve the problem. Please solve using R. Thank you!

PS = c(5.3,5,4.8,4.2,5.9,5.6,4.4,5.1,5.3,5.2,5,5,5.2,4.3)
SB = c(5.4,5.8,5.6,5.6,5.6,5.4,5,5,5.6,5.7,5.2,5.3,5.8,5.9)

Answer 1

(a)

The R code for the required confidence interval for difference of means is done in 2 methods.

R code for first method:

PS = c(5.3,5,4.8,4.2,5.9,5.6,4.4,5.1,5.3,5.2,5,5,5.2,4.3)
SB = c(5.4,5.8,5.6,5.6,5.6,5.4,5,5,5.6,5.7,5.2,5.3,5.8,5.9)
t.test(PS,SB,conf.level = 0.98,var.equal=T)$conf

Output screen:

[1] -0.8403912 -0.1024659
attr(,"conf.level")
[1] 0.98

So using first method ,98% Confidence interval for the given mean difference is :

(-0.8404 , -0.1025). (upto 4 decimal places)

R code for second method:

PS = c(5.3,5,4.8,4.2,5.9,5.6,4.4,5.1,5.3,5.2,5,5,5.2,4.3)
SB = c(5.4,5.8,5.6,5.6,5.6,5.4,5,5,5.6,5.7,5.2,5.3,5.8,5.9)
a=mean(PS)-mean(SB)
b=(sqrt((13*var(PS)+13*var(SB))/26))*(sqrt((1/14)+(1/14)))
z=qt(.99,26)
a-b*z
a+b*z
Output Screen:

[1] -0.8403912
[1] -0.1024659

So using second method ,98% Confidence interval for the given mean difference is :

(-0.8404 , -0.1025). (upto 4 decimal places)

(b) Based on the confidence interval,we can say with 98% confidence that bread treated with the preservative Potassium Sorbet grows mold faster than bread treated with Sodium Benzoate, because both the lower limit and the upper limit of the CI is negative. Also because the average amount of time it takes for bread with the preservative Potassium Sorbet to grow mold is less than the average amount of time it takes for bread with the preservative Sodium Benzoate to grow mold.