Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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:
[tex]\mu = 7.5, \sigma = 1.2[/tex]
a. What proportion of cold sufferers experience fewer than 4 days of symptoms?
This is the pvalue of Z when X = 4. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{4 - 7.5}{1.2}[/tex]
[tex]Z = -2.92[/tex]
[tex]Z = -2.92[/tex] 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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{10 - 7.5}{1.2}[/tex]
[tex]Z = 2.08[/tex]
[tex]Z = 2.08[/tex] has a pvalue of 0.9812.
X = 7
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{7 - 7.5}{1.2}[/tex]
[tex]Z = -0.42[/tex]
[tex]Z = -0.42[/tex] 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.
