Respuesta :
Answer:
The range of sample means that defines the middle 95% of the distribution of sample means is between 42.16 and 57.84.
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 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 = 50, \sigma = 20, n = 25, s = \frac{20}{\sqrt{25}} = 4[/tex]
Find the range of sample means that defines the middle 95% of the distribution of sample means.
This is from the 50 - 95/2 = 2.5th percentile to the 50 + 95/2 = 97.5th percentile.
2.5th percentile
value of X when Z has a pvalue of 0.025. So X when Z = -1.96.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-1.96 = \frac{X - 50}{4}[/tex]
[tex]X - 50 = -1.96*4[/tex]
[tex]X = 42.16[/tex]
97.5th percentile
value of X when Z has a pvalue of 0.975. So X when Z = 1.96.
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]1.96 = \frac{X - 50}{4}[/tex]
[tex]X - 50 = 1.96*4[/tex]
[tex]X = 57.84[/tex]
The range of sample means that defines the middle 95% of the distribution of sample means is between 42.16 and 57.84.
