Respuesta :
Answer:
92.65% probability that the total weight in a random sample of 47 American adults exceeds 7500 pounds in 2005.
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.
For sums the theorem can also be used, with mean [tex]n*\mu[/tex] and standard deviation [tex]s = \sqrt{n}*\sigma[/tex]
In this problem, we have that:
[tex]n = 47, \mu = 47*167 = 7849, s = \sqrt{47}*35 = 240[/tex]
Use this information to estimate the probability that the total weight in a random sample of 47 American adults exceeds 7500 pounds in 2005.
This is 1 subtracted by the pvalue of Z when X = 7500. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{7500 - 7849}{240}[/tex]
[tex]Z = -1.45[/tex]
[tex]Z = -1.45[/tex] has a pvalue of 0.0735
1 - 0.0735 = 0.9265
92.65% probability that the total weight in a random sample of 47 American adults exceeds 7500 pounds in 2005.
