Respuesta :
Answer:
0.98 = 98% probability that the average midterm score of these students is at most 75 points.
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 normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, 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.
The average midterm score of students in a certain course is 70 points.
This means that [tex]\mu = 70[/tex]
29 students are randomly selected and the standard deviation of their scores is found to be 13.15 points.
This means that [tex]\sigma = 13.15, n = 29, s = \frac{13.15}{\sqrt{29}} = 2.44[/tex]
Find the probability that the average midterm score of these students is at most 75 points.
This is the pvalue of Z when X = 75. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{75 - 70}{2.44}[/tex]
[tex]Z = 2.05[/tex]
[tex]Z = 2.05[/tex] has a pvalue of 0.98.
0.98 = 98% probability that the average midterm score of these students is at most 75 points.
