Question

the random variable x is the crew size of a randomly selected shuttle mission. its probability distribution is shown below. complete parts a through c.

x	p(x = x)
3	0.009
4	0.074
5	0.367
6	0.094
7	0.356
8	0.035

a. find and interpret the mean of the random variable.
μ = 5.624 (round to three decimal places as needed.)
interpret the mean. choose the correct answer below.
○ a. the observed value of the random variable will be less than the mean of the random variable in most observations.
○ b. as the number of observations, n, increases, the mean of the observations will approach the mean of the random variable.
○ c. as the number of observations, n, decreases, the mean of the observations will approach the mean of the random variable.
○ d. the observed value of the random variable will be equal to the mean of the random variable in most observations.

b. obtain the standard deviation of the random variable.
σ = □ (round to three decimal places as needed.)

Explanation:

Step1: Calculate the standard deviation

The formula for the standard deviation \(\sigma\) of a discrete random variable \(X\) is \(\sigma=\sqrt{\sum(x - \mu)^{2}P(X = x)}\).
First, we know \(\mu = 5.624\).
For \(x = 2\), \((2 - 5.624)^{2}\times0.065=( - 3.624)^{2}\times0.065 = 13.133\times0.065 = 0.8536\)
For \(x = 3\), \((3 - 5.624)^{2}\times0.009=( - 2.624)^{2}\times0.009=6.885\times0.009 = 0.06196\)
For \(x = 4\), \((4 - 5.624)^{2}\times0.074=( - 1.624)^{2}\times0.074 = 2.637\times0.074=0.1951\)
For \(x = 5\), \((5 - 5.624)^{2}\times0.367=( - 0.624)^{2}\times0.367=0.3894\times0.367 = 0.1430\)
For \(x = 6\), \((6 - 5.624)^{2}\times0.094=(0.376)^{2}\times0.094 = 0.1414\times0.094=0.01339\)
For \(x = 7\), \((7 - 5.624)^{2}\times0.356=(1.376)^{2}\times0.356 = 1.893\times0.356=0.6739\)
For \(x = 8\), \((8 - 5.624)^{2}\times0.035=(2.376)^{2}\times0.035 = 5.645\times0.035 = 0.1976\)

Sum these values: \(0.8536+0.06196 + 0.1951+0.1430+0.01339+0.6739+0.1976=2.1386\)

Then \(\sigma=\sqrt{2.1386}\approx1.462\)

Answer:

\(\sigma\approx1.462\)