Question

the data given below show the number of overtime hours worked in one week per employee. use the data to complete parts (a) and (b)
overtime hours 0 1 2 3 4 5 6
employees 8 14 34 57 41 30 15
(a) construct a probability distribution.
(b) graph the probability distribution using a histogram and describe its shape.
(a) construct the probability distribution by completing the table below
x 0 1 2 3 4 5 6
p(x) 0.040 0.070 0.171 0.286 0.206 0.151 0.075
(round to three decimal places as needed.)
(b) choose the correct graph of the probability distribution.
a. histogram image
b. histogram image
c. histogram image
d. histogram image

Explanation:

Step1: Calculate Total Employees

First, sum the number of employees for each overtime hour.
Total employees = \( 8 + 14 + 34 + 57 + 41 + 30 + 15 = 199 \) (Wait, wait, let's check again: 8 (0h) +14(1h)=22; +34(2h)=56; +57(3h)=113; +41(4h)=154; +30(5h)=184; +15(6h)=199. Wait, but the given \( P(x) \) values: for x=0, \( 8/199 \approx 0.040 \), x=1: \( 14/199 \approx 0.070 \), x=2: \( 34/199 \approx 0.171 \), x=3: \( 57/199 \approx 0.286 \), x=4: \( 41/199 \approx 0.206 \), x=5: \( 30/199 \approx 0.151 \), x=6: \( 15/199 \approx 0.075 \). So the probabilities are correct.

Step2: Analyze Histogram Shape

The probability distribution has \( P(x) \) values: 0.040 (x=0), 0.070 (x=1), 0.171 (x=2), 0.286 (x=3), 0.206 (x=4), 0.151 (x=5), 0.075 (x=6). So the probabilities increase from x=0 to x=3, then decrease from x=3 to x=6. So the histogram should have a peak at x=3, increasing before and decreasing after. Let's check the graphs:

• Graph A: Maybe symmetric? No, our data peaks at 3, then decreases. Wait, let's see the options. Wait, the user's graph options: Let's assume the correct graph is the one that matches the probability distribution. The probabilities for x=0:0.04, x=1:0.07, x=2:0.17, x=3:0.29, x=4:0.21, x=5:0.15, x=6:0.08. So the heights (probabilities) should be: x=0: ~0.04, x=1: ~0.07, x=2: ~0.17, x=3: ~0.29, x=4: ~0.21, x=5: ~0.15, x=6: ~0.08. So the bar for x=3 is the tallest, then x=4 is shorter than x=3, x=2 shorter than x=3, etc. So looking at the graphs, let's assume Graph A (or the one with peak at 3, increasing then decreasing). Wait, maybe the correct graph is the one that has the highest bar at x=3, then decreasing. Let's check the options. Suppose the correct graph is A (or the one that matches the probability distribution). But maybe the user's graph A has the peak at x=3. Alternatively, let's recheck the probability values.

Wait, the problem (b) is to choose the correct graph. Let's see the probability distribution:

x: 0, P=0.040

x:1, P=0.070

x:2, P=0.171

x:3, P=0.286 (highest)

x:4, P=0.206 (lower than x=3)

x:5, P=0.151 (lower than x=4)

x:6, P=0.075 (lower than x=5)

So the bars should be:

x=0: short

x=1: taller than x=0

x=2: taller than x=1

x=3: tallest

x=4: shorter than x=3

x=5: shorter than x=4

x=6: shorter than x=5

So the histogram should have a single peak at x=3, increasing from x=0 to x=3, then decreasing from x=3 to x=6. So the correct graph is the one that shows this. Let's assume the correct graph is A (or the one labeled as the one with peak at 3). But maybe the user's graph A is the correct one. Wait, maybe the answer is Graph A (or the specific one, but since the user's options are A, B, C, D, and based on the probability distribution, the correct graph is the one with the bar heights matching the probabilities. So the correct graph is the one where x=3 has the highest bar, x=2 is lower than x=3, x=4 is lower than x=3, etc.

But maybe the correct graph is A (assuming the options: Let's proceed.

For part (a), the probability distribution is already given (rounded to three decimals), so that's correct.

For part (b), the correct graph is the one that matches the probability distribution, which has a peak at x=3, increasing before and decreasing after. So the correct graph is, say, Graph A (depending on the actual options, but based on the probability values, the correct graph is the one with bars: x=0: ~0.04, x=1: ~0.07, x=2: ~0.17, x=3: ~0.29, x=4: ~0.21, x=5: ~0.15, x=6: ~0.08. So the correct graph is the one that displays this.

Answer:

(a) The probability distribution table is already completed as:

x	0	1	2	3	4	5	6

(b) The correct graph is the one that shows the probability distribution with a peak at \( x = 3 \), increasing from \( x = 0 \) to \( x = 3 \), and decreasing from \( x = 3 \) to \( x = 6 \). (Assuming the correct option is, e.g., A. [Graph A] or the specific one matching the bar heights.)