Respuesta :
Answer:
a) 22.36% probability that a Dutch male is shorter than 175 cm
b) 12.71% probability that a Dutch male is taller than 195 cm
c) 65.78% probability that a Dutch male is between 173 and 193 cm
d) We should expect 251.4 to be taller than 190 cm
Step-by-step explanation:
When the distribution is normal, we use 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 question:
[tex]\mu = 183, \sigma = 10.5[/tex]
A. What is the probability that a Dutch male is shorter than 175 cm?
This is the pvalue of Z when X = 175. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{175 - 183}{10.5}[/tex]
[tex]Z = -0.76[/tex]
[tex]Z = -0.76[/tex] has a pvalue of 0.2236
22.36% probability that a Dutch male is shorter than 175 cm.
B. What is the probability that a Dutch male is taller than 195 cm?
This is 1 subtracted by the pvalue of Z when X = 195. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{195 - 183}{10.5}[/tex]
[tex]Z = 1.14[/tex]
[tex]Z = 1.14[/tex] has a pvalue of 0.8729
1 - 0.8729 = 0.1271
12.71% probability that a Dutch male is taller than 195 cm.
C. What is the probability that a Dutch male is between 173 and 193 cm?
This is the pvalue of Z when X = 193 subtracted by the pvalue of Z when X = 173. So
X = 193
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{193 - 183}{10.5}[/tex]
[tex]Z = 0.95[/tex]
[tex]Z = 0.95[/tex] has a pvalue of 0.8289
X = 173
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{173 - 183}{10.5}[/tex]
[tex]Z = -0.95[/tex]
[tex]Z = -0.95[/tex] has a pvalue of 0.1711
0.8289 - 0.1711 = 0.6578
65.78% probability that a Dutch male is between 173 and 193 cm.
D. Out of a random sample of 1000 Dutch men, how many would we expect to be taller than 190 cm?
Proportion taller than 190cm is 1 subtracted by the pvalue of Z when X = 190. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{190 - 183}{10.5}[/tex]
[tex]Z = 0.67[/tex]
[tex]Z = 0.67[/tex] has a pvalue of 0.7486
1 - 0.7486 = 0.2514
Out of 1000:
0.2514*1000 = 251.4.
We should expect 251.4 to be taller than 190 cm
