Respuesta :
Answer:
96.33% probability that the sample average will be less than $8.
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 [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
In this problem, we have that:
[tex]\mu = 7.65, \sigma = 2.15, n = 121, s = \frac{2.15}{\sqrt{121}} = 0.1955[/tex]
What is the probability that the sample average will be less than $8?
This is the pvalue of Z when X = 8. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{8 - 7.65}{0.1955}[/tex]
[tex]Z = 1.79[/tex]
[tex]Z = 1.79[/tex] has a pvalue of 0.9633.
96.33% probability that the sample average will be less than $8.
