Question

6. Working out Choose a person aged 19 to 25 years at random and ask, “In the past seven days, how many times did you go to an exercise or fitness center or work out?” Call the response Y for short. Based on a large sample survey, here is a probability model for the answer you will get:

Days	0	1	2	3	4	5	6	7
Probability	0.68	0.05	0.07	0.08	0.05	0.04	0.01	0.02

(a) Show that this is a legitimate probability distribution.
(b) Make a histogram of the probability distribution. Describe what you see.
(c) Describe the event $Y<7$words. Wthat is$P(Y<7)$?
(d) Express the event “worked out at least once” in terms of $Y\ge 1$. What is the probability of this event?

Short answer

(a) The probability distribution provided is a legitimate probability distribution.

(b) The distribution is right-skewed, with the highest bar to the left and a tail of bars to the right.

(c) The probability is$P(Y<7)=0.98$

(d) The probability is$P(Y\ge 1)=0.32$

Step-by-step solution

Part (a) Step 1: Given information

To show that the probability distribution is a legitimate probability distribution.

Part (a) Step 2: Explanation 

Determine the sum of probabilities as follows:

Sum of probabilities $=0.68+0.05+0.08+0.05+0.01+0.02$

=1

The probability range from 0 to ,1 and the sum of the probabilities equals 1.
As a result, the probability distribution provided is a legitimate probability distribution.

Part (b) Step 1: Given information

To make a histogram of the probability distribution and interpret it.

Part (b) Step 2: Explanation

To histogram of the probability distribution as follows:

The distribution is right-skewed, with the highest bar to the left and a tail of bars to the right.

Part (c) Step 1: Given information

To describe the event $Y<7$and$P(Y<7)$.

Part (c) Step 2: Explanation

Since,$X<7$ denotes that number of days that people workout is less than 7.
Let, determine $P(X<7)$as follows:

$P(X<7)=P(X=1)+P(X=2)+\dots ..+P(X=6)$

$$\begin{gathered} =0.68+0.05+\dots ..+0.01 \\ =0.98 \end{gathered}$$

As a result, the probability is 0.98

Part (d) Step 1: Given information

To determine the probability of the event. And to express the event “worked out at least once” in terms of Y.

Part (d) Step 2: Explanation

In terms of Y, the event "exercise at least once" may be expressed as $Y\ge 1$

Then $P(Y\ge 1)$

$P(Y\ge 1)=1-P(Y<0)$

$=1-0.68$

$=0.32$

As a result, the probability is 0.32.