Question

A generic brand of all-purpose fertilizer G claims to contain 10 percent phosphorous, a nutrient which helps plants grow strong root systems. A leading name brand of all-purpose fertilizer N advertises having a phosphorous concentration at least 30 percent higher than the generic brand. The most important consideration is that the mean phosphorous concentration in each brand is at least as high as advertised; a slightly higher phosphorous concentration is not concerning, but having too little phosphorous will hurt or slow the growth of plants’ root systems. Further, brand N is significantly more expensive than brand G per pound, so consumers who purchase brand N fertilizer expect the product to produce stronger root systems as advertised.

You work for a consumer protection group that has received complaints about both brands. The complainants claim that the mean phosphorous concentration in each fertilizer is lower than advertised. Additionally, you wish to determine whether brand N’s claim of having a phosphorous concentration at least 30% higher than brand N is true. If any of the companies’ claims are found to be untrue, the consumer protection group will bring a class-action lawsuit against the brand or brands responsible.
The consumer protection group bought bags of each brand of fertilizer and sent them to a lab for analysis. The phosphorous content (as a percent) for each bag, along with a column indicating which brand produced the bag is detailed below:

Phosphorous (% per bag), Brand
9.79522878032876, G
10.9302191178489, G
12.1732852582726, N
9.0544640624081, G
10.0475170197499, N
11.6656993826563, N
12.6642653530907, N
9.10468690374451, G
9.03436435420168, G
13.7352205503801, N
14.5489736025146, N
9.77948772403382, N
12.2986303191169, N
10.3807530975497, G
12.1950434195649, G
9.36377060086138, G
9.35399703787208, G
11.6263999062187, N
14.8368893022518, N
10.3076684543705, G
8.95153287738328, G
9.38069973700502, G
12.2352561489847, N
8.75328753808385, G
7.67158718250212, G
11.060128497079, N
13.3621542490895, N
12.1468548485131, N
9.32726632730324, N
12.2110173028274, N
9.41454096948369, G
10.4809518447645, N
9.96123587641246, G
10.1635577800051, G
10.8408606925903, N
16.1557438655007, N
7.94990087762832, N
9.44666693523413, G
9.16255880009391, G
10.8115193560222, G

a) What is the appropriate hypothesis test to determine whether brand G has the mean phosphorous concentration advertised?
b) What is the appropriate hypothesis test to determine whether brand N has the mean phosphorous concentration advertised?
c) What is the appropriate hypothesis test to determine whether brand N has a mean phosphorous concentration at least 30% higher than brand G as advertised?

Answer 1

Let and be the mean phosphorous concentration of generic beand G and leading name brand N.

a)

The appropriate hypothesis test to determine whether brand G has the mean phosphorous concentration advertised is,

The test is that the mean phosphorous concentration advertised is at least 10%. So, this is a right tailed test (one tailed test).

Since, we do not know the population standard deviation of phosphorous concentration in both brands, we would use t test.

b)

Brand of all-purpose fertilizer N advertises having a phosphorous concentration at least 30 percent higher than the generic brand (which is 10%)

The appropriate hypothesis test to determine whether brand N has the mean phosphorous concentration advertised

The test is that the mean phosphorous concentration advertised is at least 40%. So, this is a right tailed test (one tailed test).

Since, we do not know the population standard deviation of phosphorous concentration in both brands, we would use t test.

c)

The appropriate hypothesis test to determine whether brand N has a mean phosphorous concentration at least 30% higher than brand G as advertised is,

The test is that the mean difference in phosphorous concentration advertised is at least 10%. So, this is a right tailed test (one tailed test).

Since, we do not know the population standard deviation of phosphorous concentration in both brands, we would use t test.