can someone please help me solve this?
the u.s weather bureau has provided the following information about the total annual rainfall (in inches) in hawaii for the years 1965 to 1984.
15 12 31.5 17 11.2 13 12.4 16.3 14.6 10 12.8 14.1 14.7 10.7 15.2 10.9 18.4 12 15.9 10
a) use Chebyshev's theorem to find an interval centered about the mean for the annual rain fall in which you would expect at least 88.9% of the years to fall.
b)use Chebyshev's theorem to find an interval centered about the mean for the annual rain fall in which you would expect at least 75% of the years to fall.
c)for the data given above find the median and the coefficent of variation.
I will try to keep this simple add all the rainfall numbers up (287.7) divide by how many parts you had (20)=14.385 this is your mean.
Make a chart with three columns the first will be each individual rainfall amount. Second column will be each amount with the mean subtracted from it. Third column will be the number in column 2 squared.
ex:
15 .615 .378
12 -2.385 5.688
31.5 17.115 292
17 2.615 6.838
do this for all 20.
add up column 3 you should get 417.1855
now you divide that total (417.1855) by n-1 (20-1=19) 417.1855/19=21.957 this is your variance. take the square root of the variance and you get standard deviation (4.686).
Finally part of the answer take standard deviation/mean to get coeffecient of variation (4.686/14.385= .325756) so 32.57%
Cheby just puts bounds on how much data must lie close to the mean. Here comes the whole 1-(1/k^2). k has to be 3.3333 to equal 88.9% and k has to be 2 to equal 75%.
Now we take the mean (14.385)and subtract k (3.3333)*standard dev(4.686)
14.385-15.61998=-1.235 to get the bottom of the interval. to get the top add 3.3333*standard dev to the mean 14.385+15.61998=30.005.
the 75 % question is just making k=2
14.385- (2)(4.686)=5.013
14.385+ (2)(4.686)=23.757.
to get the medain sort all your dat in order 10, 10, 10.7, 10.9, 11.2, 12, 12, 12.4,12.8, 13, 14, 14.6, 14.7,15, 15.2, 15.9, 16.3, 17, 18.4, 31.5. cross off the two end numbers until you get to the middle 10 gone 31.5 gone 10 gone 18.4 gone etc... if you have an odd number that middle one is the median in this case you have an even amount so you take the average of 13 and 14 to get13.5. Hope this helped
To solve this problem and answer the questions, you will need to follow these steps:
Step 1: Calculate the mean
- Add up all the rainfall numbers: 15 + 12 + 31.5 + 17 + 11.2 + 13 + 12.4 + 16.3 + 14.6 + 10 + 12.8 + 14.1 + 14.7 + 10.7 + 15.2 + 10.9 + 18.4 + 12 + 15.9 + 10 = 287.7
- Divide the sum by the number of parts you had (20): 287.7 / 20 = 14.385
- The mean for the annual rainfall is 14.385 inches.
Step 2: Calculate the variance and standard deviation
- Make a chart with three columns: the first column for each individual rainfall amount, the second column for each amount with the mean subtracted from it, and the third column for the squared values of the amounts in the second column.
- Calculate the squared difference for each amount with the mean subtracted from it.
- Add up the values in the third column.
- Divide the total by n-1 (n is the number of data points, in this case, 20-1 = 19).
- Take the square root of the result to get the standard deviation.
- In this case, the variance is 417.1855 and the standard deviation is 4.686.
Step 3: Calculate the coefficient of variation
- Divide the standard deviation by the mean to get the coefficient of variation.
- In this case, the coefficient of variation is 4.686 / 14.385 = 0.325756 or 32.57%.
Step 4: Calculate the intervals using Chebyshev's theorem
- Chebyshev's theorem states that at least (1 - (1 / k^2)) of the data falls within k standard deviations of the mean.
- For the first question, where at least 88.9% of the years are expected to fall within the interval, set k = 3.3333 (obtained from solving 1 - (1 / k^2) = 0.889).
- Subtract k (3.3333) multiplied by the standard deviation (4.686) from the mean (14.385) to get the bottom of the interval and add k multiplied by the standard deviation to the mean to get the top of the interval.
- For the second question, where at least 75% of the years are expected to fall within the interval, set k = 2 (obtained from solving 1 - (1 / k^2) = 0.75).
- Perform the same calculations as in the previous step.
- The interval for the first question is (-1.235, 30.005) and the interval for the second question is (5.013, 23.757).
Step 5: Calculate the median
- Sort all the data in ascending order: 10, 10, 10.7, 10.9, 11.2, 12, 12, 12.4, 12.8, 13, 14, 14.6, 14.7, 15, 15.2, 15.9, 16.3, 17, 18.4, 31.5.
- Cross off the two end numbers until you reach the middle. In this case, cross off 10 and 31.5, then 10 and 18.4, and so on, until you have two numbers left.
- Since you have an even number of data points, take the average of the two middle numbers: (13 + 14) / 2 = 13.5.
- The median for the data is 13.5.
I hope this explanation helps you to solve the problem and provides a clear understanding of the steps involved.