Respuesta :
Answer:
There is a 2.28% probability that the average weight of these 64 adult males is over 205 pounds.
Step-by-step explanation:
To solve this question, we have to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
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.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s= \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 197, \sigma = 32, n = 64, s = \frac{32}{\sqrt{64}} = 4[/tex]
What is the probability that the average weight of these 64 adult males is over 205 pounds?
This is 1 subtracted by the pvalue of Z when X = 205.
So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{205 - 197}{4}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772
1 - 0.9772 = 0.0228
There is a 2.28% probability that the average weight of these 64 adult males is over 205 pounds.
