The following table gives the total caseload in the New York State courts form 2004 through 2009.
Year 2004 2005 2006 2007 2008 2009
Cases (in MM) 4.2 4.3 4.6 4.5 4.7 4.7
A) Find the mean and standard deviation of the data.
B) Consider the second table for the state of Pennsylvania
Year 2004 2005 2006 2007 2008 2009
Cases (in MM) 3.4 3.5 3.2 3.7 3.9 3.8
Find the mean and standard deviation of the second table
C) What does a comparison of the means say about the number of cases?
a) Find mean. To do this I made a table to include the cases as probability distributions.
Frequencies: 4.2 appears once, 4.3 appears once, 4.6 appears once, 4.5 appears once, 4.7 appears twice. 6 total items.
X 4.2 4.3 4.6 4.5 4.7
1/6 1/6 1/6 1/6 2/6
So then the mean is computed like this:
(4.2) * (1/6) + (4.3) * (1/6) + (4.6) * (1/6) + (4.5)* (1/6) + (4.7) * (2/6)
= 4.5
standard deviation :
set up like this ? (4.2)*(1/6-4.5)^2+(4.3)*(1/6-4.5)^2+(4.6)*(1/6-4.5)^2+(4.5)*(1/6-4.5)^2+(4.7)*(2/6-4.5)^2
Take the answer and square root it.(I think it is 20 but I maybe made a mistake or didn't round).
For the second table:
Frequencies: 3.4 appears once, 3.5 appears once, 3.2 appears once, 3.7 appears once, 3.9 appears once, 3.8 appears once. 6 total items.
X 3.4 3.5 3.2 3.7 3.9 3.8
p(x=x) 1/6 1/6 1/6 1/6 1/6
(3.4) * (1/6) + (3.5) * (1/6)+(3.2) * (1/6) + (3.7) * (1/6)+ (3.9) * (1/6) + (3.8) * (1/6) = 3.6 = mean
standard deviation =
(3.4)*(1/6-3.6)^2+(3.5)*(1/6-3.5)^2+(3.2)*(1/6-3.6)^2+(3.7)*(1/6-3.6)^2+(3.9)*(1/6-3.6)^2+(3.8)*(1/6-3.6)
square the answer and i think it might be 16 but please correct me if i'm wrong.
For part c, it looks like the mean is larger for NY so I think NY has more court cases than Pennsylvania.
New York:
mean = (4.2 + 4.3 + 4.6 + 4.5 + 4.7 + 4.7)/6 = 4.55
squares of differences:
(4.55-4.2)^2 = .1225
(4.55-4.3)^2 = .0625
(4.55-4.6)^2 = .0025
(4.55-4.5)^2 = .0025
(4.55-4.7)^2 = .0225
(4.55-4.7)^2 = .0225
sd = √(sum of above/6) = 19685..
I carried all decimals in my calculator, leave it up to you to round off if needed
repeat for Penns.
To find the mean of the given data for New York State courts, you correctly set up a probability distribution table. The frequencies you listed are correct, but the probabilities should be the frequencies divided by the total number of items in the data, which is 6.
The correct probability distribution table for the New York data is as follows:
X | 4.2 | 4.3 | 4.6 | 4.5 | 4.7 |
P(X) | 1/6 | 1/6 | 1/6 | 1/6 | 2/6 |
To find the mean, you multiply each value by its corresponding probability, and sum them up:
Mean = (4.2)*(1/6) + (4.3)*(1/6) + (4.6)*(1/6) + (4.5)*(1/6) + (4.7)*(2/6)
= 4.5
So the mean of the New York data is 4.5.
Moving on to the standard deviation calculation, you are on the right track. The formula to calculate the standard deviation is:
Standard deviation = √(Σ[(X - mean)^2 * P(X)])
For each value in the New York data, you need to subtract the mean, square the difference, multiply by the corresponding probability, and sum them up. Then take the square root of the result.
Standard deviation = √[(4.2 - 4.5)^2 * (1/6) + (4.3 - 4.5)^2 * (1/6) + (4.6 - 4.5)^2 * (1/6) + (4.5 - 4.5)^2 * (1/6) + (4.7 - 4.5)^2 * (2/6)]
After simplifying and calculating, the standard deviation for the New York data is approximately 0.1414.
Now, to find the mean and standard deviation for the second table (Pennsylvania), follow the same steps using the data provided:
X | 3.4 | 3.5 | 3.2 | 3.7 | 3.9 | 3.8 |
P(X) | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |
Mean = (3.4)*(1/6) + (3.5)*(1/6) + (3.2)*(1/6) + (3.7)*(1/6) + (3.9)*(1/6) + (3.8)*(1/6)
Similarly, calculate the standard deviation using the formula mentioned earlier.
Now, for the comparison of the means between the two datasets, you can observe that the mean for New York (4.5) is higher than the mean for Pennsylvania (3.5). This indicates that, on average, the total caseload in New York State courts was higher than in Pennsylvania during the given years.