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Answer :
Answer:
a) 58.32% probability that his weight will be greater than 164 pounds.
b) 76.11% probability that 12 randomly selected people will have a neam that is greater than 164 pounds.
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 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 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.
In this problem, we have that:
[tex]\mu = 171, \sigma = 34[/tex]
a. find the probability that if a person is randomly selected, his weight will be greater than 164 pounds.
This is 1 subtracted by the pvalue of Z when X = 164. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{164 - 171}{34}[/tex]
[tex]Z = -0.21[/tex]
[tex]Z = -0.21[/tex] has a pvalue of 0.4168
1 - 0.4168 = 0.5832
58.32% probability that his weight will be greater than 164 pounds.
b. Find the probability that 12 randomly selected people will have a neam that is greater than 164 pounds.
Now [tex]n = 12, s = \frac{34}{\sqrt{12}} = 9.81[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{164 - 171}{9.81}[/tex]
[tex]Z = -0.71[/tex]
[tex]Z = -0.71[/tex] has a pvalue of 0.2389
1 - 0.2389 = 0.7611
76.11% probability that 12 randomly selected people will have a neam that is greater than 164 pounds.
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