Respuesta :
Answer:
11.90% probability that the mean profit for the sample was between 0 million and 5.1 million
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 question, we have that:
[tex]\mu = 6.54, \sigma = 10.45, n = 73, s = \frac{10.45}{\sqrt{73}} = 1.2231[/tex]
In a random sample of 73 companies from the NYSE, what is the probability that the mean profit for the sample was between 0 million and 5.1 million?
This is the pvalue of Z when X = 5.1 subtracted by the pvalue of Z when X = 0. So
X = 5.1
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{5.1 - 6.54}{1.2231}[/tex]
[tex]Z = -1.18[/tex]
[tex]Z = -1.18[/tex] has a pvalue of 0.1190
X = 0
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0 - 6.54}{1.2231}[/tex]
[tex]Z = -5.35[/tex]
[tex]Z = -5.35[/tex] has a pvalue of 0
0.1190 - 0 = 0.1190
11.90% probability that the mean profit for the sample was between 0 million and 5.1 million
