a The distribution is valid if it has the following two properties.
The probability of each value of X is greater than or equal to 0 and less than or equal to 1.
The sum of all probabilities of X is equal to 1.
Let's look at the given table.
Number of Stores
Probability
0 - 2
0.35
3 - 5
0.32
6 - 8
0.17
9 - 11
0.11
12+
0.05
Let X be the number of stores visited. We can see that the probability of every value of X is greater than 0 and less than 1. Now, let's see if the sum of all the probabilities is equal to 1.
0.35+0.32+0.17+0.11+0.05=1
Since both requirements for a valid distribution are met, we can see that this is a valid distribution.
b We want to find the probability that a randomly shopper will shop at more than 5 stores, but less than 12. This is the same as finding the probability that the value of X when X is greater than 5 and less than 12.
P(5<X<12)=?
Let's look at the give table.
Number of Stores
Probability
X=0 - 2
0.35
X=3 - 5
0.32
X=6 - 8
0.17
X=9 - 11
0.11
X=12+
0.05
We can see that the rows with X=6 - 8 and X=9 - 11 satisfy these conditions. Our event is made up of 2 simpler events, so this is a compound event. Since the 2 events are mutually exclusive events, we can calculate P(5<X<12) by adding the probabilities that each event occurs.
The probability that a randomly chosen shopper will shop at more than 5 and less than 12 stores is 0.28.
c We will make a bar graph of the data. Each bar is associated with the number of stores visited by shoppers. The height of each bar will show the probability of the event associated with the bar.
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