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) 0.37 0.09 0.05 0.14 0.35
(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.
mean = x*p(x)
a) 10(.37) + 11(.09) + 12(.05) + 13(.14) + 14(.35)
3.7 + 0.99 + 0.6 + 1.82 + 4.9 = 12.01
x^2*p(x)
variance = 10^2 (.37) + 11^2 (0.99) + 12^2 (0.05) +13 ^2 (0.14) +14^2 (0.35)
37 + 119.79 + 7.2 + 23.66 + 68.6 = 256.25
Standard deviation = sqrt(variance - xbar^2)
=sqrt(256.25 - 12.01^2))
= sqrt(256.25- 144.2401)
= sqrt(112.0099)
= 10.58
(a) To find the mean of the number of ships that arrive at a harbor on a given day, we need to calculate the weighted average of the values.
Mean (µ) = Σ(x * P(x))
= (10 * 0.37) + (11 * 0.09) + (12 * 0.05) + (13 * 0.14) + (14 * 0.35)
= 3.70 + 0.99 + 0.60 + 1.82 + 4.90
= 12.01
Therefore, the mean number of ships that arrive at a harbor on a given day is approximately 12.01.
(b) To find the standard deviation (σ) of the number of ships that arrive at a harbor on a given day, we can use the formula:
σ = √(Σ(x^2 * P(x)) - µ^2)
= √((10^2 * 0.37) + (11^2 * 0.09) + (12^2 * 0.05) + (13^2 * 0.14) + (14^2 * 0.35) - 12.01^2)
= √((100 * 0.37) + (121 * 0.09) + (144 * 0.05) + (169 * 0.14) + (196 * 0.35) - 144.2401)
= √(37 + 10.89 + 7.2 + 23.66 + 68.6 - 144.2401)
= √(147.35 - 144.2401)
= √3.1099
= 1.76
Therefore, the standard deviation of the number of ships that arrive at a harbor on a given day is approximately 1.76.
(a) To find the mean of the number of ships that arrive at a harbor on a given day, we need to find the expected value. We can calculate this by multiplying each value of x by its corresponding probability and summing them up.
Mean (μ) = (10 x 0.37) + (11 x 0.09) + (12 x 0.05) + (13 x 0.14) + (14 x 0.35)
= 3.7 + 0.99 + 0.6 + 1.82 + 4.9
= 11.01
Therefore, the mean number of ships that arrive at a harbor on a given day is 11.01.
(b) To find the standard deviation (σ) of the number of ships that arrive at a harbor on a given day, we need to calculate the variance first. The variance is the sum of the squared differences of each value from the mean, multiplied by their respective probabilities.
Var (σ^2) = [(10 - 11.01)^2 x 0.37] + [(11 - 11.01)^2 x 0.09] + [(12 - 11.01)^2 x 0.05] + [(13 - 11.01)^2 x 0.14] + [(14 - 11.01)^2 x 0.35]
= [(-1.01)^2 x 0.37] + [(-0.01)^2 x 0.09] + [(0.99)^2 x 0.05] + [(1.99)^2 x 0.14] + [(2.99)^2 x 0.35]
= [1.0201 x 0.37] + [0.0001 x 0.09] + [0.9801 x 0.05] + [3.9601 x 0.14] + [8.9401 x 0.35]
= 0.377037 + 0.000009 + 0.049005 + 0.554414 + 3.124135
= 4.1046
Now, we can calculate the standard deviation by taking the square root of the variance.
σ = √4.1046
= 2.03
Therefore, the standard deviation of the number of ships that arrive at a harbor on a given day is approximately 2.03.
To find the mean of the number of ships that arrive at a harbor on a given day, we need to calculate the weighted average.
(a) Mean (µ) Calculation:
Multiply each value of x by its corresponding probability (P(x)) and sum them up:
Mean (µ) = 10 * 0.37 + 11 * 0.09 + 12 * 0.05 + 13 * 0.14 + 14 * 0.35
Mean (µ) = 3.7 + 0.99 + 0.6 + 1.82 + 4.9
Mean (µ) = 11.01
So, the mean number of ships that arrive at the harbor on a given day is 11.01.
To find the standard deviation (σ) of the number of ships that arrive at a harbor on a given day, we need to calculate the variance and then take its square root.
(b) Standard Deviation (σ) Calculation:
Variance (σ^2) is calculated by subtracting the mean squared from the expected value of squaring each value and its corresponding probability:
Variance (σ^2) = [(10^2) * 0.37 + (11^2) * 0.09 + (12^2) * 0.05 + (13^2) * 0.14 + (14^2) * 0.35] - (11.01^2)
Variance (σ^2) = [37 * 0.37 + 121 * 0.09 + 144 * 0.05 + 169 * 0.14 + 196 * 0.35] - (121.2101)
Variance (σ^2) = [13.69 + 10.89 + 7.20 + 23.66 + 68.60] - (121.2101)
Variance (σ^2) = 123.04 - 121.2101
Variance (σ^2) = 1.8299
Therefore, the variance is 1.8299.
To find the standard deviation, take the square root of the variance:
Standard Deviation (σ) = √(1.8299)
Standard Deviation (σ) ≈ 1.35
So, the standard deviation of the number of ships that arrive at the harbor on a given day is approximately 1.35.