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.
12.01 is this answer I got
(b) Find the standard deviation, ó, of the number of ships that arrive at a harbor on a given day. and I got 19.19
First step:
Check to see if it is a discrete probability distribution by summing all probabilities. If it does not equal one, either the given distribution is incomplete, or it is not a discrete distribution.
Second step:
Mean (expected value of X, E(X), μ)
=ΣX*P(X)
=10*0.37+11*0.09....+14*0.35
=12.01 (so your answer is correct)
Third step:
Note that for a discrete distribution, the standard deviation will never be outside the domain of the distribution. 19.19 is outside the domain, so it cannot be right.
Variance σ²= Σ(X-μ)² * P(X)
=(10-12.01)²*0.37+...(14-12.01)²*0.35
=3.1 (approx.)
Standard deviation
=√(σ²)
=1.8 (approx.)
I came up with standard deviation of 2.88 when I worked it out like you have it laid out. But is that right???? I also got 10.58 and that is not right. I have one more chance at this one...
Check your calculations!
Did you get standard deviation or σ² as 2.88?
Don't understand how you got 2.88 and 10.58. Can you post what you did?
For the variance, I have
1.49+0.09+0.00+0.14+1.39=3.1
To find the mean of the number of ships that arrive at a harbor on a given day, you need to multiply each possible value by its respective probability and then sum them up.
(a) Mean (µ) calculation:
10 * 0.37 + 11 * 0.09 + 12 * 0.05 + 13 * 0.14 + 14 * 0.35 = 3.7 + 0.99 + 0.6 + 1.82 + 4.9 = 11.01
So, the mean of the number of ships that arrive at a 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, you need to calculate the deviation of each possible value from the mean, square each deviation, multiply the squared deviation by its respective probability, sum them up, and take the square root of the result.
(b) Standard deviation (σ) calculation:
[(10 - 11.01)^2 * 0.37] + [(11 - 11.01)^2 * 0.09] + [(12 - 11.01)^2 * 0.05] + [(13 - 11.01)^2 * 0.14] + [(14 - 11.01)^2 * 0.35] = [1.0369 * 0.37] + [0.0001 * 0.09] + [0.0001 * 0.05] + [3.9601 * 0.14] + [6.2401 * 0.35] = 0.3836 + 0.000009 + 0.000005 + 0.5544 + 2.184035 = 3.122044
Now, take the square root of the result:
Square root of 3.122044 = 1.7666
So, the standard deviation (σ) of the number of ships that arrive at a harbor on a given day is approximately 1.77 (rounded to two decimal places).