Respuesta :
Answer:
a) 85.31% probability that 1 randomly selected adult male has a weight greater than 148 lb.
b) 100% probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb.
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 = 189, \sigma = 39[/tex]
a.) Find the probability that 1 randomly selected adult male has a weight greater than 148 lb.
This is 1 subtracted by the pvalue of Z when X = 148.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{148 - 189}{39}[/tex]
[tex]Z = -1.05[/tex]
[tex]Z = -1.05[/tex] has a pvalue of 0.1469
1 - 0.1469 = 0.8531
85.31% probability that 1 randomly selected adult male has a weight greater than 148 lb.
b.) Find the probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb.
Now [tex]n = 27, s = \frac{39}{\sqrt{27}} = 7.51[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{148 - 189}{7.51}[/tex]
[tex]Z = -5.46[/tex]
[tex]Z = -4.56[/tex] has a pvalue of 0
1 - 0 = 1
100% probability that a sample of 27 randomly selected adult males has a mean weight greater than 148 lb.
