Two sides of a right triangle measure 5 inches and 6 inches, and the third side measures√61 inches.The length of the third side
2 answers:
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Answer:
0.8254 = 82.54% probability that the mean of this sample is between 19.25 hours and 21.0 hours
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 normal distributions can be solved using the z-score formula.
In a set with mean and standard deviation
, the z-score 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 p-value, 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 and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Mean of 20 hours, standard deviation of 6:
This means that
Sample of 150:
This means that
What is the probability that the mean of this sample is between 19.25 hours and 21.0 hours?
This is the p-value of Z when X = 21 subtracted by the p-value of Z when X = 19.5. So
X = 21
By the Central Limit Theorem
has a p-value of 0.9793
X = 19.5
has a p-value of 0.1539
0.9793 - 0.1539 = 0.8254
0.8254 = 82.54% probability that the mean of this sample is between 19.25 hours and 21.0 hours
Note that 
in the given closed interval, so the area is exactly given by the definite integral
(no absolute values needed!)
Integrating gives
(no absolute values needed!)
Integrating gives