In the following probability distribution, the random variable X represents the number of marriages an individual aged 15 years or older has been involved in. Complete parts (a) through (f).
x & 0 & 1 & 2 & 3 & 4 & 5 \
\hline $P ( x )$ & 0.291 & 0.521 & 0.148 & 0.033 & 0.005 & 0.002
(c) Compute and interpret the mean of the random variable X.
The mean is 0.946 marriages.
(Type an integer or a decimal. Do not round.)
Which of the following interpretations of the mean is correct?
A. As the number of experiments increases, the mean of the observations will approach the mean of the random variable.
B. The observed value of an experiment will be equal to the mean of the random variable in most experiments.
C. As the number of experiments decreases, the mean of the observations will approach the mean of the random variable.
D. The observed value of an experiment will be less than the mean of the random variable in most experiments.
(d) Compute the standard deviation of the random variable X.
The standard deviation is $\square$ marriages.
(Round to one decimal place as needed.)
Answer (2)
The standard deviation of the random variable X is approximately 0.9 marriages. The correct interpretation of the mean is that as the number of experiments increases, the mean of the observations will approach the mean of the random variable (Option A).
;
The problem asks for the standard deviation of a random variable given its probability distribution and mean.
First, calculate the variance using the formula: Va r ( X ) = ∑ [ x 2 ∗ P ( x )] − m e an ( X ) 2 = 0.645084 .
Then, find the standard deviation by taking the square root of the variance: S D ( X ) = 0.645084 ≈ 0.803 .
Round the standard deviation to one decimal place: 0.8 .
Explanation
Understand the problem and provided data We are given the probability distribution for the random variable X, which represents the number of marriages an individual aged 15 years or older has been involved in. We are also given the mean of X, which is 0.946. Our goal is to compute the standard deviation of X and choose the correct interpretation of the mean.
Identify the correct interpretation of the mean The correct interpretation of the mean is: As the number of experiments increases, the mean of the observations will approach the mean of the random variable. This corresponds to option A.
Calculate the variance To calculate the standard deviation, we first need to calculate the variance. The formula for variance is: Va r ( X ) = ∑ [ x 2 ∗ P ( x )] − m e an ( X ) 2 We have the values of x and P(x), and we know the mean is 0.946. So, we can plug in the values: Va r ( X ) = [ 0 2 ∗ 0.291 + 1 2 ∗ 0.521 + 2 2 ∗ 0.148 + 3 2 ∗ 0.033 + 4 2 ∗ 0.005 + 5 2 ∗ 0.002 ] − ( 0.946 ) 2 Va r ( X ) = [ 0 + 0.521 + 4 ∗ 0.148 + 9 ∗ 0.033 + 16 ∗ 0.005 + 25 ∗ 0.002 ] − 0.894916 Va r ( X ) = [ 0.521 + 0.592 + 0.297 + 0.08 + 0.05 ] − 0.894916 Va r ( X ) = 1.54 − 0.894916 Va r ( X ) = 0.645084
Calculate the standard deviation Now, we calculate the standard deviation by taking the square root of the variance: S D ( X ) = Va r ( X ) S D ( X ) = 0.645084 S D ( X ) ≈ 0.8031712146236317 Rounding to one decimal place, we get 0.8.
Final Answer The standard deviation of the random variable X is approximately 0.8 marriages.
Examples
Understanding the distribution of marriages can be useful in social sciences for demographic studies, policy making related to family services, and for businesses targeting specific marital status groups. For example, insurance companies might use this data to assess risk, or marketing firms might tailor campaigns based on the likelihood of individuals being married.
