Respuesta :
Answer:
14.46% probability that the instructor can finish grading in 4 and a half hours
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.
For sums of n values from a distribution, the mean is [tex]n\mu[/tex] and the standard deviation is [tex]s = \sqrt{n}\sigma[/tex]
In this problem, we have that:
[tex]\mu = 6*50 = 300, s = \sqrt{50}*4 = 28.2843[/tex]
What is the probability that the instructor can finish grading in 4 and a half hours
Four and half hours is 4.5*60 = 270.
So this probability is the pvalue of Z when X = 270.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{270 - 300}{28.2843}[/tex]
[tex]Z = -1.06[/tex]
[tex]Z = -1.06[/tex] has a pvalue of 0.1446
14.46% probability that the instructor can finish grading in 4 and a half hours
