Question

S) Suppose that a local second hand store is selling grab bags for $20 each in order to get rid of excess stock. They have created a total of 300 grab bags, most of which contain various items worth $15 in total. However, there are 40 grab bags which contain items worth $40 in total, 20 grab bags worth $60 in total, and a single grab bath worth $120 in total. Suppose you purchase one grab bag. Let X be the monetary value of a grab bag

Answer 1

Explanation:

Hello!

a)

The variable of interest is

X: monetary value of a grab bag. ($)

The possible values this variable can take are:

X: 15; 40; 60; 120

To calculate the relative frequency of each possible value of the variable you have to divide the number of times this value is observed by the total of observations (sample size)

There where a total of 300 bags created, from these, 40 cost $40, 20 cost $60, 1 cost $120, and the rest have a monetary value of $15.

There are then 300- (40+20+1)= 239 bags with a monetary value of $15

The frequencies for each value of X are:

X= 15 ⇒ p(X)= 239/300= 0.797

X= 40 ⇒ p(X)= 40/300= 0.133

X= 60 ⇒ p(X)= 20/300=0.067

X= 120⇒ p(X)= 1/300= 0.003

The probability distribution for X is:

Xi: 15; 40; 60; 120

p(Xi):0.797; 0.133; 0.067; 0.003

∑p(Xi)= 0.797+0.133+0.067+0.003=1

b)

To calculate the mean you have to use the following formula:

E(X)= ∑Xp(X)= 15*0.797+40*0.133+60*0.067+120*0.003= 21.655≅ $21.7

c)

The standard deviation is the square root of the variance:

Variance:

V(X)= ∑X²p(X)-(∑Xp(X))²= 676.525 - (21.655)²= 207.585

∑X²p(X)= 15²*0.797+40²*0.133+60²*0.067+120²*0.003= 676.525

Standard deviation:

√V(X)= √207.585= $14.407= $14.41

d)

Considered the probability distribution, it is more likely to buy a $15 worth bag than any of the others, it is therefore not worthwhile to spend $20 in a bag full of merchandise worth $15.

I hope this helps!