Respuesta :
Answer:
0.962 = 96.2% probability that a simple random sample of 100 adult males from this county has a mean weight between 172 and 188 lbs.
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 establishes 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.
The distribution of weights of adult males in a certain county is strongly right-skewed with a mean weight of 185 lbs and standard deviation 16 lbs.
This means that [tex]\mu = 185, \sigma = 16[/tex]
Sample of 100:
This means that [tex]n = 100, s = \frac{16}{\sqrt{100}} = 1.6[/tex]
What is the probability that a simple random sample of 100 adult males from this county has a mean weight between 172 and 188 lbs?
This is the p-value of Z when X = 188 subtracted by the p-value of Z when X = 172. So
X = 188
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{188 - 185}{1.6}[/tex]
[tex]Z = 1.875[/tex]
[tex]Z = 1.875[/tex] has a p-value of 0.9620
X = 172
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{172 - 185}{1.6}[/tex]
[tex]Z = -8.125[/tex]
[tex]Z = -8.125[/tex] has a p-value of 0.
0.9620 - 0 = 0.962
0.962 = 96.2% probability that a simple random sample of 100 adult males from this county has a mean weight between 172 and 188 lbs.
