JT EspanolYou roll one die. What is the probability that you roll a 6?
2 answers:
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Answer:
1. 0.7421 = 74.21% probability the elevator is overloaded.
2. D.No, there is a good chance that 10 randomly selected people will exceed the elevator capacity.
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 establishes 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.
Assume that weights of males are normally distributed with a mean of 166 lb and a standard deviation of 29 lb.
This means that
Sample of 10.
This means that
1.The probability the elevator is overloaded is?
Probability that the sample mean is above 160 pounds, which is 1 subtracted by the p-value of Z when X = 160. So
By the Central Limit Theorem
has a p-value of 0.2579.
1 - 0.2579 = 0.7421
0.7421 = 74.21% probability the elevator is overloaded.
2. Does this elevator appear to be safe?
High probability of the elevator being overloaded, so not safe. Correct answer is given by option D.
Answer:
And if we solve for a we got
So the value of height that separates the bottom 80% of data from the top 20% is 250.31.
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the room hotel rate of a population, and for this case we know the distribution for X is given by:
Where and
For this part we want to find a value a, such that we satisfy this condition:
(a)
(b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.80 of the area on the left and 0.20 of the area on the right it's z=0.842. On this case P(Z<0.842)=0.8 and P(z>0.842)=0.2
If we use condition (b) from previous we have this:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got
So the value of height that separates the bottom 80% of data from the top 20% is 250.31.