Respuesta :
Answer:
27.88% probability that a random sample of 5 has a mean between 677 and 693.
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 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.
In this problem, we have that:
[tex]\mu = 675, \sigma = 44, n = 5, s = \frac{44}{\sqrt{5}} = 19.6774[/tex]
The probability that a random sample of 5 has a mean between 677 and 693.
This is the pvalue of Z when X = 693 subtracted by the pvalue of Z when X = 677. So
X = 693
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{693 - 675}{19.6774}[/tex]
[tex]Z = 0.91[/tex]
[tex]Z = 0.91[/tex] has a pvalue of 0.8186
X = 677
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{677 - 675}{19.6774}[/tex]
[tex]Z = 0.1[/tex]
[tex]Z = 0.1[/tex] has a pvalue of 0.5398
0.8186 - 0.5398 = 0.2788
27.88% probability that a random sample of 5 has a mean between 677 and 693.
