Question

It is said that sufferers of a cold virus experience symptoms for 7 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 experience fewer than 4 days of symptoms?
b. What proportion of cold sufferers experience symptoms for between 7 and 10 days?

Answer 1

Answer:

a) 0.18% of cold sufferers experience fewer than 4 days of symptoms

b) 64.40% of cold sufferers experience symptoms for between 7 and 10 days.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 7.5, \sigma = 1.2$

a. What proportion of cold sufferers experience fewer than 4 days of symptoms?

This is the pvalue of Z when X = 4. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{4 - 7.5}{1.2}$

$Z = -2.92$

$Z = -2.92$ has a pvalue of 0.0018.

So 0.18% of cold sufferers experience fewer than 4 days of symptoms.

b. What proportion of cold sufferers experience symptoms for between 7 and 10 days?

This is the pvalue of Z when X = 10 subtracted by the pvalue of Z when X = 7. So

X = 10

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{10 - 7.5}{1.2}$

$Z = 2.08$

$Z = 2.08$ has a pvalue of 0.9812.

X = 7

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{7 - 7.5}{1.2}$

$Z = -0.42$

$Z = -0.42$ has a pvalue of 0.3372.

So 0.9812 - 0.3372 = 0.644 = 64.40% of cold sufferers experience symptoms for between 7 and 10 days.