Respuesta :
Answer:
50% probability of a sample mean being less than 12,751 or greater than 12,754
Step-by-step explanation:
To solve this question, we have 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 random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean \mu and standard deviation, which is also called standard error [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 12751, \sigma = 2.1, n = 37, s = \frac{2.1}{\sqrt{37}} = 0.3456[/tex]
Find the probability of a sample mean being less than 12,751 or greater than 12,754
Less than 12,751
pvalue of Z when X = 12751.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{12751 - 12751}{0.3456}[/tex]
[tex]Z = 0[/tex]
[tex]Z = 0[/tex] has a pvalue of 0.5.
50% probability of the sample mean being less than 12,751.
Greater than 12,754
1 subtracted by the pvalue of Z when X = 12,754.
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{12754 - 12751}{0.3456}[/tex]
[tex]Z = 8.68[/tex]
[tex]Z = 8.68[/tex] has a pvalue of 1
1 - 1 = 0
0% probability of the sample mean being greater than 12754
Less than 12,751 or greater than 12,754
50 + 0 =50
50% probability of a sample mean being less than 12,751 or greater than 12,754
