The number of peanuts in a 16-ounce can of nut munchies is normally distributed with a mean of 96.3 and a standard deviation of 2.4 peanuts. the number of peanuts in a 20-ounce can of gone nuts is normally distributed with a mean of 112.6 and a standard deviation of 2.8 peanuts.
(a) carmen purchased a 16-ounce can of nut munchies and counted 100 peanuts.
what is the z-score for this can of peanuts?
(b) angelo purchased a 20-ounce can of gone nuts and counted 116 peanuts.
what is the z-score for this can of peanuts?
(c) carmen declares that purchasing her can of nut munchies with 100 peanuts is less likely than angelo purchasing a can of gone nuts with 116 peanuts. is carmen’s statement correct? use the definition of a z-score to support or refute carmen’s claim.

To calculate for the z-score we use the formula:

z=(x-μ)/σ

thus the answers to questions will be as follows:

Step-by-step explanation:

a]Carmen purchased a 16-ounce can of Nut Munchies and counted 100 peanuts. What is the z-score for this can of peanuts?

x=100

μ=96.3

σ=2.4

z=(100-96.3)/2.4

z=1.542

b]Angelo purchased a 20-ounce can of Gone Nuts and counted 116 peanuts. What is the z-score for this can of peanuts?

x=116

μ=112.6

σ=2.8

thus

z=(116-112.6)/2.8

z=1.214

c]Carmen declares that purchasing her can of Nut Munchies with 100 peanuts is less likely than Angelo purchasing a can of Gone Nuts with 116 peanuts. Is Carmen’s statement correct? Use the definition of a z-score to support or refute Carmen’s claim.

This is very correct because because by definition of z-score, Munchies with 100 peanuts is 1.542 away from the mean as compared to Munchies with 116 peanuts which is 1.214 standard deviations from the mean hence the higher likelihood. 

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!