Respuesta :
Answer:
53.84% probability that for a group of 10 students, the mean number of times they go to the movies each year is between 14 and 18 times.
Step-by-step explanation:
To solve this problem, 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], a sample of 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 = 17, \sigma = 8, n = 10, s = \frac{8}{\sqrt{10}} = 2.53[/tex]
What is the probability that for a group of 10 students, the mean number of times they go to the movies each year is between 14 and 18 times?
This probability is the pvalue of Z when X = 18 subtracted by the pvalue of Z when X = 14. So
X = 18
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{18 - 17}{2.53}[/tex]
[tex]Z = 0.4[/tex]
[tex]Z = 0.4[/tex] has a pvalue of 0.6554
X = 14
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{14 - 17}{2.53}[/tex]
[tex]Z = -1.19[/tex]
[tex]Z = -1.19[/tex] has a pvalue of 0.1170
0.6554 - 0.1170 = 0.5384
53.84% probability that for a group of 10 students, the mean number of times they go to the movies each year is between 14 and 18 times.
