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.