Respuesta :
Answer:
58.32% probability that a randomly selected application will report a GMAT score of less than 600
93.51% probability that a sample of 50 randomly selected applications will report an average GMAT score of less than 600
98.38% probability that a sample of 100 randomly selected applications will report an average GMAT score of less than 600
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 = 591, \sigma = 42[/tex]
What is the probability that a randomly selected application will report a GMAT score of less than 600?
This is the pvalue of Z when X = 600. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{600 - 591}{42}[/tex]
[tex]Z = 0.21[/tex]
[tex]Z = 0.21[/tex] has a pvalue of 0.5832
58.32% probability that a randomly selected application will report a GMAT score of less than 600
What is the probability that a sample of 50 randomly selected applications will report an average GMAT score of less than 600?
Now we have [tex]n = 50, s = \frac{42}{\sqrt{50}} = 5.94[/tex]
This is the pvalue of Z when X = 600. So
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{600 - 591}{5.94}[/tex]
[tex]Z = 1.515[/tex]
[tex]Z = 1.515[/tex] has a pvalue of 0.9351
93.51% probability that a sample of 50 randomly selected applications will report an average GMAT score of less than 600
What is the probability that a sample of 100 randomly selected applications will report an average GMAT score of less than 600?
Now we have [tex]n = 50, s = \frac{42}{\sqrt{100}} = 4.2[/tex]
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{600 - 591}{4.2}[/tex]
[tex]Z = 2.14[/tex]
[tex]Z = 2.14[/tex] has a pvalue of 0.9838
98.38% probability that a sample of 100 randomly selected applications will report an average GMAT score of less than 600
