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Jorjalilli8081
04.02.2020 •
Mathematics
A28-year-old man pays $181 for a one-year life insurance policy with coverage of $150,000. if the probability that he will live through the year is 0.9994, what is the expected value for the insurance policy?
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Ответ:
Ответ:
0.985 = 98.5% probability that the sample mean will be between $7.75 and $8.20.
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.
The average price for a movie in the United States in 2012 was $7.96. Assume the population st. dev. is $0.50.
This means that![\mu = 7.96, \sigma = 0.5](/tpl/images/1393/6534/4948a.png)
Sample of 30:
This means that![n = 30, s = \frac{0.5}{\sqrt{30}}](/tpl/images/1393/6534/d7f6e.png)
What is the probability that the sample mean will be between $7.75 and $8.20?
This is the p-value of Z when X = 8.2 subtracted by the p-value of Z when X = 7.75.
X = 8.2
By the Central Limit Theorem
X = 7.75
0.9957 - 0.0157 = 0.985
0.985 = 98.5% probability that the sample mean will be between $7.75 and $8.20.