Respuesta :
Answer:
P25 = 205.365cm
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 random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], 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]
In this problem, we have that:
[tex]\mu = 205.6, \sigma = 1.1, n = 10, s = \frac{1.1}{\sqrt{10}} = 0.34785[/tex]
Find P25, which is the average length separating the smallest 25% bundles from the largest 75% bundles. P25
This is the value of X when Z has a pvalue of 0.25. So X when Z = -0.675.
So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-0.675 = \frac{X - 205.6}{0.34785}[/tex]
[tex]X - 205.6 = -0.675*0.34785[/tex]
[tex]X = 205.365[/tex]
P25 = 205.365cm
