The number of ships to arrive at a harbor on any given day is a random variable represented by x. The probability distribution of x is as follows. (Give your answers correct to two decimal places.)
x 10 11 12 13 14
P(x) 0.38 0.1 0.07 0.12 0.33
(a) Find the mean of the number of ships that arrive at a harbor on a given day.
(b) Find the standard deviation, ó, of the number of ships that arrive at a harbor on a given day.
(a) Finding the mean:
To find the mean, we need to multiply each value of x by its corresponding probability, then sum up all the results.
Mean (µ) = (10 * 0.38) + (11 * 0.1) + (12 * 0.07) + (13 * 0.12) + (14 * 0.33)
Mean (µ) = 3.8 + 1.1 + 0.84 + 1.56 + 4.62
Mean (µ) = 11.92
Therefore, the mean number of ships that arrive at the harbor on a given day is 11.92.
(b) Finding the standard deviation:
The formula for the standard deviation (σ) is as follows:
σ = sqrt(∑(x - µ)^2 * P(x))
To calculate it, we need to subtract the mean from each value of x, square the result, multiply it by the corresponding probability, sum up all the results, and then take the square root.
First, let's calculate ∑(x - µ)^2 * P(x):
(10 - 11.92)^2 * 0.38 + (11 - 11.92)^2 * 0.1 + (12 - 11.92)^2 * 0.07 + (13 - 11.92)^2 * 0.12 + (14 - 11.92)^2 * 0.33
(1.92)^2 * 0.38 + (-0.92)^2 * 0.1 + (-0.08)^2 * 0.07 + (1.08)^2 * 0.12 + (2.08)^2 * 0.33
7.3728 * 0.38 + 0.8464 * 0.1 + 0.0064 * 0.07 + 1.1664 * 0.12 + 4.3232 * 0.33
2.800064 + 0.08464 + 0.000448 + 0.139968 + 1.426256
σ^2 ≈ 4.451376
σ = sqrt(4.451376)
σ ≈ 2.11
Therefore, the standard deviation of the number of ships that arrive at the harbor on a given day is approximately 2.11.
To find the mean of the number of ships that arrive at a harbor on a given day, you will need to multiply each value of x by its corresponding probability and then sum them up.
(a) Mean (μ) = ∑(x * P(x))
10 * 0.38 = 3.80
11 * 0.10 = 1.10
12 * 0.07 = 0.84
13 * 0.12 = 1.56
14 * 0.33 = 4.62
μ = 3.80 + 1.10 + 0.84 + 1.56 + 4.62 = 11.92
Therefore, the mean number of ships that arrive at a harbor on a given day is 11.92.
Now, to find the standard deviation (σ), you will need to calculate the variance first.
(b) Variance (σ²) = ∑((x - μ)² * P(x))
(10 - 11.92)² * 0.38 = 1.4556 * 0.38 = 0.5527
(11 - 11.92)² * 0.10 = 0.0084 * 0.10 = 0.0008
(12 - 11.92)² * 0.07 = 0.0006 * 0.07 = 0.0000
(13 - 11.92)² * 0.12 = 1.4192 * 0.12 = 0.1703
(14 - 11.92)² * 0.33 = 4.2204 * 0.33 = 1.3927
σ² = 0.5527 + 0.0008 + 0.0000 + 0.1703 + 1.3927 = 2.1165
Finally, to find the standard deviation, take the square root of the variance:
σ = √2.1165 ≈ 1.45
Therefore, the standard deviation of the number of ships that arrive at a harbor on a given day is approximately 1.45.
To find the mean (also known as the expected value) of the number of ships that arrive at a harbor on a given day:
(a) Multiply each value of x by its corresponding probability P(x):
(10 * 0.38) + (11 * 0.1) + (12 * 0.07) + (13 * 0.12) + (14 * 0.33)
Then, add up the results of those multiplications:
3.8 + 1.1 + 0.84 + 1.56 + 4.62
Finally, compute the sum:
11.92
Therefore, the mean number of ships that arrive at a harbor on a given day is 11.92.
To find the standard deviation (ó) of the number of ships that arrive at a harbor on a given day:
(b) Subtract the mean from each value of x and square the result:
(10 - 11.92)^2, (11 - 11.92)^2, (12 - 11.92)^2, (13 - 11.92)^2, (14 - 11.92)^2
Then, multiply each squared result by its corresponding probability P(x):
0.38 * (10 - 11.92)^2, 0.1 * (11 - 11.92)^2, 0.07 * (12 - 11.92)^2, 0.12 * (13 - 11.92)^2, 0.33 * (14 - 11.92)^2
Compute the sum of all the squared and multiplied results:
0.38 * 2.7264, 0.1 * 0.0084, 0.07 * 0.0056, 0.12 * 0.0304, 0.33 * 4.5424
Calculate each value:
1.034432, 0.00084, 0.000392, 0.003648, 1.498592
Add up the squared and multiplied results:
1.538904
Finally, take the square root of the sum:
√(1.538904)
Therefore, the standard deviation (ó) of the number of ships that arrive at a harbor on a given day is approximately 1.24 (rounded to two decimal places).