Answer :
The amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents is $680 approx. The probability a random person from Superior has less than $400 or more than $1000 in their bank account is 0.9247
How to get the z scores?
If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have
[tex]X \sim N(\mu, \sigma)[/tex]
(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex])
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write
[tex]P(Z \leq z) = P(Z < z) )[/tex]
Also, know that if we look for Z = z in z tables, the p-value we get is
[tex]P(Z \leq z) = \rm p \: value[/tex]
Let we take:
X = The balance of a person’s bank account in Superior, WI
Then, by the given data, we have:
[tex]X \sim N(\mu = 580, \sigma = 125)[/tex]
Let the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents be X = x dollars. This is the money, below which lie 80% of Superior, WI residents, and so as their income, or values of X.
Symbolically, we have:
[tex]P(X < x) = 80\% = 0.8[/tex]
Converting X to standard normal variate Z, we have:
[tex]Z = \dfrac{X - \mu}{\sigma} = \dfrac{X - 580}{125}\\[/tex]
Thus, the probability statement can be rewritten as:
[tex]P(X < x) =0.8\\P\left( Z < z = \dfrac{x -580}{125} \right) = 0.8[/tex]
From the z-tables, the value of Z for which the p-value comes out as between 0.84 and 0.85, let it be their average 0.845.
Thus, we get:
[tex]P(Z < z = 0.845) \approx 0.8[/tex]
Thus, we get:
[tex]z = \dfrac{x - 580}{125} \approx 0.8\\\\x \approx 0.8 \times 125 + 580 = 680[/tex]
Thus, the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents is $680 approx.
Now, for second question, we need:
The probability a random person from Superior has less than $400 or more than $1000 in their bank account.
So, income of a random person from Superior is a random value of X.
So, this probability is written as:
[tex]P(400 < X < 1000)[/tex]
Rewritting it, we get:
[tex]P(400 < X < 1000) = P(X < 1000) - P(X \leq 400)[/tex]
Converting X to Z (standard normal variate), we get:
[tex]P(400 < X < 1000) = P(X < 1000) - P(X \leq 400)\\\\P(400 < X < 1000) = P\left(Z < \dfrac{1000-580}{125} \right) - P\left(Z \leq \dfrac{400-580}{125} \right)\\\\P(400 < X < 1000) = P( Z \leq 3.36) - P(Z \leq -1.44)[/tex]
The p-value for Z = 3.36 is 0.9996
The p-value for Z = -1.44 is 0.0749
Thus, we get:
[tex]P(400 < X < 1000) = P( Z \leq 3.36) - P(Z \leq -1.44)\\\\P(400 < X < 1000) = 0.9996 - 0.0749 = 0.9247[/tex]
Thus, the probability a random person from Superior has less than $400 or more than $1000 in their bank account is 0.9247
Thus, the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents is $680 approx. The probability a random person from Superior has less than $400 or more than $1000 in their bank account is 0.9247
Learn more about z-score here:
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