Respuesta :
Answer:
61.70% approximate probability that X will be more than 0.5 away from the population mean
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 [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 36, \sigma = 6, n = 36, s = \frac{6}{\sqrt{36}} = 1[/tex]
What is the approximate probability that X will be more than 0.5 away from the population mean?
This is the probability that X is lower than 36-0.5 = 35.5 or higher than 36 + 0.5 = 36.5.
Lower than 35.5
Pvalue of Z when X = 35.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{35.5 - 36}{1}[/tex]
[tex]Z = -0.5[/tex]
[tex]Z = -0.5[/tex] has a pvalue of 0.3085.
30.85% probability that X is lower than 35.5.
Higher than 36.5
1 subtracted by the pvalue of Z when X = 36.5. SO
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{36.5 - 36}{1}[/tex]
[tex]Z = 0.5[/tex]
[tex]Z = 0.5[/tex] has a pvalue of 0.6915.
1 - 0.6915 = 0.3085
30.85% probability that X is higher than 36.5
Lower than 35.5 or higher than 36.5
2*30.85 = 61.70
61.70% approximate probability that X will be more than 0.5 away from the population mean
