Question

On the midnight shift, the number of patients with head trauma in an emergency room has the probability distribution shown below.

x	0	1	2	3	4	5	Total
P(x)	.04	.33	.27	.21	.13	.02	1.00

(a) Calculate the mean and standard deviation. (Round your mean value to 2 decimal places and standard deviation to 3 decimal places.)

Mean
Standard deviation

Answer 1

Answer 2

To calculate the mean and standard deviation, we need to perform the following calculations:

1. Calculate the mean (expected value): Mean (μ) = Σ (x * P(x))

2. Calculate the variance (σ^2): Variance (σ^2) = Σ [(x - μ)^2 * P(x)]

3. Calculate the standard deviation (σ): Standard Deviation (σ) = √σ^2

where x is the number of patients with head trauma and P(x) is the probability of having x patients.

Let's perform the calculations step by step:

x | 0 | 1 | 2 | 3 | 4 | 5 |

P(x) | 0.04| 0.33| 0.27| 0.21| 0.13| 0.02|

1. Calculate the mean (expected value): Mean (μ) = (0 * 0.04) + (1 * 0.33) + (2 * 0.27) + (3 * 0.21) + (4 * 0.13) + (5 * 0.02) = 0.00 + 0.33 + 0.54 + 0.63 + 0.52 + 0.10 = 2.12

2. Calculate the variance (σ^2): Variance (σ^2) = [(0 - 2.12)^2 * 0.04] + [(1 - 2.12)^2 * 0.33] + [(2 - 2.12)^2 * 0.27] + [(3 - 2.12)^2 * 0.21] + [(4 - 2.12)^2 * 0.13] + [(5 - 2.12)^2 * 0.02] = [(-2.12)^2 * 0.04] + [(-1.12)^2 * 0.33] + [(-0.12)^2 * 0.27] + [(0.88)^2 * 0.21] + [(1.88)^2 * 0.13] + [(2.88)^2 * 0.02] = 0.1792 + 0.4007 + 0.0095 + 0.1605 + 0.4559 + 0.2333 = 1.4391

3. Calculate the standard deviation (σ): Standard Deviation (σ) = √σ^2 = √1.4391 ≈ 1.199

Therefore, the mean (μ) is approximately 2.12 and the standard deviation (σ) is approximately 1.199.

Answer 3

To calculate the mean and standard deviation of the probability distribution, we'll use the following formulas:

Mean (μ) = Σ (x * P(x)) Standard Deviation (σ) = √(Σ ((x - μ)^2 * P(x)))

where: x = Number of patients with head trauma P(x) = Probability of having x number of patients with head trauma Σ = Summation (the sum is taken over all x values)

Given the probability distribution:

x	0	1	2	3	4	5
P(x)	0.04	0.33	0.27	0.21	0.13	0.02

Let's calculate the mean first:

Mean (μ) = (0 * 0.04) + (1 * 0.33) + (2 * 0.27) + (3 * 0.21) + (4 * 0.13) + (5 * 0.02) Mean (μ) = 0 + 0.33 + 0.54 + 0.63 + 0.52 + 0.10 Mean (μ) = 2.12 (rounded to 2 decimal places)

Now, let's calculate the standard deviation:

σ^2 = [(0 - 2.12)^2 * 0.04] + [(1 - 2.12)^2 * 0.33] + [(2 - 2.12)^2 * 0.27] + [(3 - 2.12)^2 * 0.21] + [(4 - 2.12)^2 * 0.13] + [(5 - 2.12)^2 * 0.02]

σ^2 = [(-2.12)^2 * 0.04] + [(-1.12)^2 * 0.33] + [(-0.12)^2 * 0.27] + [(0.88)^2 * 0.21] + [(1.88)^2 * 0.13] + [(2.88)^2 * 0.02]

σ^2 = [4.4944 * 0.04] + [1.2544 * 0.33] + [0.0144 * 0.27] + [0.7744 * 0.21] + [3.5344 * 0.13] + [8.2944 * 0.02]

σ^2 = 0.179776 + 0.414312 + 0.003888 + 0.162624 + 0.459072 + 0.165888 σ^2 = 1.38556

Now, calculate the standard deviation:

σ = √1.38556 σ ≈ 1.177 (rounded to 3 decimal places)

So, the mean is approximately 2.12 and the standard deviation is approximately 1.177.