Respuesta :
Answer:
0.0021 = 0.21% probability that they have a mean height greater than 72.3 inches.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, 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 normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Assume that the heights of men are normally distributed with a mean of "71.3" inches and a standard deviation of 2.1 inches.
This means that [tex]\mu = 71.3, \sigma = 2.1[/tex]
Sample of 36:
This means that [tex]n = 36, s = \frac{2.1}{\sqrt{36}} = 0.35[/tex]
Find the probability that they have a mean height greater than 72.3 inches.
This is 1 subtracted by the pvalue of Z when X = 72.3. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{72.3 - 71.3}{0.35}[/tex]
[tex]Z = 2.86[/tex]
[tex]Z = 2.86[/tex] has a pvalue of 0.9979
1 - 0.9979 = 0.0021
0.0021 = 0.21% probability that they have a mean height greater than 72.3 inches.
