Question

The administration department of PIMS Hospital Islamabad surveyed the number of days 400 randomly chosen patients stayed in the hospital after an operation.
The data are:
Days (of hospital say): 1-3 4-6 7-9 10-12 13-15 16-18 19-21 22-24
Frequency: 36 18 88 42 18 18 8 10
a) Calculate the standard deviation and mean.
b) According to Chebyshev’s theorem, how many stays should be between 0 and 17 days? How many are actually in that interval?
c) How many stays can we expect between 0 and 17 days?

Answer 1

Answer on Question #45579 – Math – Statistics and Probability

Problem.

The administration department of PIMS Hospital Islamabad surveyed the number of days 400 randomly chosen patients stayed in the hospital after an operation.

The data are:

Days (of hospital say): 1-3 4-6 7-9 10-12 13-15 16-18 19-21 22-24

Frequency: 36 18 88 42 18 18 8 10

a) Calculate the standard deviation and mean.

b) According to Chebyshev's theorem, how many stays should be between 0 and 17 days? How many are actually in that interval?

c) How many stays can we expect between 0 and 17 days?

Remark:

I suppose that there is mistake in a problem, as $36 + 18 + 88 + 42 + 18 + 18 + 8 + 10 = 238 \neq 400$. I suppose that there the sample has size 238.

Solution:

a) From the data

$$\text{Mean} = \frac{2 \cdot 36 + 5 \cdot 18 + 8 \cdot 88 + 11 \cdot 42 + 14 \cdot 18 + 17 \cdot 18 + 20 \cdot 8 + 23 \cdot 10}{238} \approx 9.56.$$

Mean of sum of midpoints squares

$$= \frac{4 \cdot 36 + 25 \cdot 18 + 64 \cdot 88 + 121 \cdot 42 + 196 \cdot 18 + 289 \cdot 18 + 400 \cdot 8 + 529 \cdot 10}{238} \approx 119.86.$$

Standard deviation $= \mathrm{SD} = \sqrt{\text{Mean of sum of midpoints squares} - \text{Mean}^2} = 1.38$.

b) Chebyshev's theorem or Chebyshev's inequality says that $1 - \frac{1}{k^2}$ of the distribution's values are within $k$ standard deviations of the mean.

0-17 range equals Mean $\pm$ 7SD (9.56 $\pm$ 7 $\cdot$ 1.38), so we can expect $1 - \frac{1}{7^2} = 0.98 = 98\%$ of 238 or at least 233 stays.

There are actually from $36 + 18 + 88 + 42 + 18 = 202$ to $202 + 18 = 220$ stays between 0 and 17 days.

c) By the empirical rule in the range Mean $\pm$ 7SD will lie $99.99\%$ of stays (237 stays).

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