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What is the value of x? Enter your answer, as a decimal, in the box.
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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 and standard deviation
, the zscore of a measure X is given by:
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 and standard deviation
, a sample of size n can be approximated to a normal distribution with mean
and standard deviation
In this problem, we have that:
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
By the Central Limit Theorem
has a pvalue of 0.6554
X = 14
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.