Question

the following table displays a frequency distribution for the number of crew - members on each shuttle mission over a 10 - year period. let x denote the crew size on a randomly selected shuttle from this time period.
crew size: 3 4 5 6 7
frequency: 4 7 38 22 1

1. obtain the probability distribution of x.

x: 3 4 5 6 7
p(x): 0.051 0.090 0.487 0.282 0.013 (round to three decimal places as needed.)

1. construct a probability histogram for x. choose the correct histogram below.

oa. ob. oc. od.

Explanation:

Step1: Calculate total frequency

$4 + 7+38 + 12+1=62$

Step2: Calculate probabilities for each crew - size

For $X = 3$: $P(3)=\frac{4}{62}\approx0.065$
For $X = 4$: $P(4)=\frac{7}{62}\approx0.113$
For $X = 5$: $P(5)=\frac{38}{62}\approx0.613$
For $X = 6$: $P(6)=\frac{12}{62}\approx0.194$
For $X = 7$: $P(7)=\frac{1}{62}\approx0.016$

To construct the probability histogram:
The $x -$axis represents the crew - size ($X$ values: 3, 4, 5, 6, 7) and the $y -$axis represents the probability $P(X)$. The bar for $X = 3$ has height approximately 0.065, for $X = 4$ has height approximately 0.113, for $X = 5$ has height approximately 0.613, for $X = 6$ has height approximately 0.194 and for $X = 7$ has height approximately 0.016.

We would need to visually inspect the given histograms (not shown in full detail here) to choose the correct one. But if we assume the correct histogram has bars in ascending order of $X$ values (3 - 7) with heights corresponding to the calculated probabilities, we can make a selection.

Since the probabilities are $P(3)\approx0.065$, $P(4)\approx0.113$, $P(5)\approx0.613$, $P(6)\approx0.194$, $P(7)\approx0.016$, the histogram should have a relatively short bar for $X = 3$, a slightly taller one for $X = 4$, a very tall bar for $X = 5$, a moderately tall bar for $X = 6$ and a very short bar for $X = 7$.

Answer:

a.

$X$	$P(X)$
4	0.113
5	0.613
6	0.194
7	0.016

b. (No specific answer for the histogram selection as the histograms are not fully described, but the above - mentioned characteristics should be used for selection)