Respuesta :
Answer:
100% probability that the sample mean scores will be between 87 and 124 points
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 \mu and standard deviation, which is also called standard error [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 105, \sigma = 20, n = 20, s = \frac{20}{\sqrt{20}} = 4.47[/tex]
What is the probability that the sample mean scores will be between 87 and 124 points
This is the pvalue of Z when X = 124 subtracted by the pvalue of Z when X = 87. So
X = 124
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{124 - 105}{4.47}[/tex]
[tex]Z = 4.25[/tex]
[tex]Z = 4.25[/tex] has a pvalue of 1
X = 87
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{87 - 105}{4.47}[/tex]
[tex]Z = -4.25[/tex]
[tex]Z = -4.25[/tex] has a pvalue of 0
1 - 0 = 1
100% probability that the sample mean scores will be between 87 and 124 points
