# Answer to Question #25088 in Statistics and Probability for Carol Parducci

Question #25088

It is said that sufferers of a cold virus experience symptoms for seven days. However, the amount of time is actually a normally distributed random variable whose mean is 7.5 days and whose standard deviation is 1.2 days. a. What proportion of cold sufferers experiences less than 4 days of symptoms? b. What proportion of cold sufferers experiences symptoms for between 7 and 10 days?

Expert's answer

The normal distribution has probability density:

f(x) = 1/Sqrt[2 Pi]/sigma Exp[-(x -mean)^2/2/sigma^2)]

In our case

sigma = 1.2

mean = 7.5

a.& proportion of cold sufferers experiences less than 4 days of symptoms equals:

P(x<4) = Integrate[1/Sqrt[2 Pi]/1.2 Exp[-(x - 7.5)^2/2/1.2^2], {x, -Infinity, 4}] = 0.00176897 = 0.18 %

b.& proportion of cold sufferers experiences symptoms for between 7 and 10 days equals:

P(7<x<10) = Integrate[1/Sqrt[2 Pi]/1.2 Exp[-(x - 7.5)^2/2/1.2^2], {x, 7, 10}] = 0.642928 = 64.3 %

f(x) = 1/Sqrt[2 Pi]/sigma Exp[-(x -mean)^2/2/sigma^2)]

In our case

sigma = 1.2

mean = 7.5

a.& proportion of cold sufferers experiences less than 4 days of symptoms equals:

P(x<4) = Integrate[1/Sqrt[2 Pi]/1.2 Exp[-(x - 7.5)^2/2/1.2^2], {x, -Infinity, 4}] = 0.00176897 = 0.18 %

b.& proportion of cold sufferers experiences symptoms for between 7 and 10 days equals:

P(7<x<10) = Integrate[1/Sqrt[2 Pi]/1.2 Exp[-(x - 7.5)^2/2/1.2^2], {x, 7, 10}] = 0.642928 = 64.3 %

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