David B. answered • 04/16/21
Math and Statistics need not be scary
This looks like part of an exam, which isn't supposed to be answered here. I can give some hints and methods
a) If you have access to excel, make a spread sheet with one row being the days in hospital
b) The next row being the expected value. for each block (i.e. for the block 1-3 days the expected value would be 2, for block 19-21 days the expected value would be 20).
c) Add a third row to represent the counts given for each of these blocks.
d) Add a fourth row to represent the count times the expected value - this is your total patient days for each column
If you sum all the patient days by the number of patients (260) you will get the expected value (or average) patient days. (note: this is an estimate as the exact break down of patient days within the categories is not know and the sample sizes are in many cases too small to expect the CLT to be effective. Also, the variances will be even more uncertain.
| 1-3 | 4-6 | 7-9 | 10-12 | 13-15 | 16-18 | 19-21 | 22-24 |
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| 2 | 5 | 8 | 11 | 14 | 17 | 20 | 23 |. total - 260 patients
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| 32. | 108 | 67 | 28 | 14 | 7 | 3 | 1 |
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| 64 | 540 | ....... . . . . . . . . . . . . . . . . . . . . . . | 23 | total patient days
Calculating the standard deviation requires calculating the mean squared error. (remember your formula for calculating variance)
e) Calculate the error for each column (assuming all patients in each column stayed the expected value of days for that column). for the first column that would be ( 2-7.1) or -5.1
f)Square the error for each column (for column one that would be 26.01)
g) weigh the error squared for each column (for column one that would be 832.32)
h) sum the square of the errors to get the SSE.
i) divide the SSE by the number of patients (see above ). this is the MSE or mean squared error
j). MSE is the variance of the sample. To get the variance of the population based on the sample multiply the answer by n/(n-1). sample variance = 14.9172
k). oh... the question asked for the sample standard deviation - that you get from the variance.
The rest of the problem requires only simple graphing or table reading skills, although one doesn't need a bar chart to see from the data that the mode will be , by definition, 4-6 days
note: question c requires the student to make unwarranted assumptions of how many people stayed 18 days. It is 1/3 of the block 16-18 but the underlying distribution is most probably not uniform. Assuming uniformity the expected number of people staying 18 days would be 2 1/3 as a best guess. add that to the 3 for block 19-21, and 1 for block 22-24, and you get 6 1/3 people had stays more than 17 days. I strongly suspect that the original creator of the test question was looking for an answer of 4 by assuming that for each block of days all the people in the block stayed exactly the expected number of days, a really really bad assumption. Poor question.
Quartiles based on standard deffiniton assume knowledge of all the values, not summary data.
Creating the quartile deviations based on the column numbers as stated finds Q1 = column 4-6 (expected value of 5 based on individual patient days. Although this column also includes the median.
The fourth quantile(Q4) is defined as the middle number between the largest number (maximum) and the median. this would be the 7-9day column (expected value of 8) based on patient days.
The quartile deviation would be (8-5)/2 or 1.5. (remember this is a highly right skewed distribution)
This is using the data as given with only 8 outcomes with weights.
Remember, discuss definitions for terms with your instructors. Definitions are important and can differ between instructors with similar sounding terms having different definitions.
