The monthly income of residents of Daisy City is normally distributed with a mean of $3000 and a standard deviation of $500.

a. What is the random variable?
b. The mayor of Daisy City makes $2250 a month. What percentage of Daisy City's residents has incomes that are more than the mayor?
c. Individuals with incomes of less than $1985 per monthe are exempt from city taxes. What percentage of residents is exempt from city taxes.
d. What are the minimum and the maximum incomes of the middle 95% of the residents
e. Two hundred residents have incomes of at least $4440 per month. What is the population of Daisy City.

a. The random variable is the monthly income of residents of Daisy City.

b. To find the percentage of residents with incomes that are more than the mayor ($2250), we need to calculate the z-score and then use a z-table. The z-score is calculated as (2250 - 3000) / 500 = -1.5. Looking up the z-score (-1.5) in a z-table, we find that the percentage of residents with incomes greater than the mayor is 100% - 93.32% = 6.68% (approximately).
c. To find the percentage of residents exempt from city taxes, we need to calculate the z-score for an income of $1985. The z-score is (1985 - 3000) / 500 = -2.03. Looking up the z-score in a z-table, we find that the percentage of residents with incomes less than $1985 per month is 2.27% (approximately). Therefore, the percentage of residents exempt from city taxes is 100% - 2.27% = 97.73% (approximately).
d. To find the minimum and maximum incomes of the middle 95% of residents, we need to find the z-scores corresponding to the cumulative probabilities of 0.025 and 0.975. Looking up these z-scores in a z-table, we find that the z-scores are approximately ±1.96. Going back to the original formula, we can calculate the minimum income as 1.96 * 500 + 3000 = $3980, and the maximum income as -1.96 * 500 + 3000 = $2020. Therefore, the minimum and maximum incomes of the middle 95% of residents are $2020 and $3980 respectively.
e. To determine the population of Daisy City, we need to use the information that there are 200 residents with incomes of at least $4440 per month. To find the z-score for this income, we calculate (4440 - 3000) / 500 = 2.88. Looking up the z-score in a z-table, we find that the percentage of residents with incomes at least $4440 is approximately 0.9979. Therefore, the population of Daisy City can be estimated by dividing 200 by 0.9979, which gives us approximately 200.4. Since we cannot have a fraction of a person, we can estimate the population of Daisy City to be 200.

a. The random variable is the monthly income of residents of Daisy City.

b. To find the percentage of Daisy City's residents that have incomes that are more than the mayor's ($2250), we need to find the area to the right of $2250 in the normal distribution. We can calculate this using the z-score formula: z = (x - μ) / σ, where x is the value we want to find the area for, μ is the mean, and σ is the standard deviation.

z = (2250 - 3000) / 500
z = -1.5

Looking up the area to the right of -1.5 in the standard normal distribution table, we find that it is approximately 0.93319. To get the percentage, we multiply this by 100:

Percentage = 0.93319 * 100 = 93.319%

Therefore, approximately 93.319% of Daisy City's residents have incomes that are more than the mayor's.

c. To find the percentage of residents who are exempt from city taxes (incomes less than $1985), we again use the z-score formula:

z = (1985 - 3000) / 500
z = -2.03

Looking up the area to the left of -2.03 in the standard normal distribution table, we find that it is approximately 0.0219. Again, to get the percentage, we multiply this by 100:

Percentage = 0.0219 * 100 = 2.19%

Therefore, approximately 2.19% of Daisy City's residents are exempt from city taxes.

d. The middle 95% of the residents' incomes fall within two standard deviations from the mean. We can find the minimum and maximum incomes using the z-score formula:

For the minimum income, we look up the area to the left of -1.96 (the z-score corresponding to the lower end of the middle 95%) in the standard normal distribution table:

z = -1.96

The corresponding value is approximately -0.975.

Minimum income = -0.975 * 500 + 3000 = $2512.50

For the maximum income, we look up the area to the right of 1.96 (the z-score corresponding to the upper end of the middle 95%) in the standard normal distribution table:

z = 1.96

The corresponding value is approximately 0.975.

Maximum income = 0.975 * 500 + 3000 = $3487.50

Therefore, the minimum and maximum incomes of the middle 95% of the residents are $2512.50 and $3487.50, respectively.

e. To find the population of Daisy City, we can use the information given. We know that 200 residents have incomes of at least $4440 per month. To find the population, we need to find the z-score for $4440 and then use the formula:

z = (x - μ) / σ

z = (4440 - 3000) / 500
z = 2.88

Looking up the area to the left of 2.88 in the standard normal distribution table, we find that it is approximately 0.9985. This means that approximately 99.85% of the population has incomes lower than $4440.

Let's assume the population of Daisy City is P.

Number of residents with incomes lower than $4440 = P * 0.9985
Number of residents with incomes equal to or greater than $4440 = 200

According to the information given, the number of residents with incomes equal to or greater than $4440 is 200. So:

200 = P * (1 - 0.9985)
200 = P * 0.0015

Solving for P:

P = 200 / 0.0015 ≈ 133,333.33

Therefore, the population of Daisy City is approximately 133,333.

a. The random variable in this scenario is the monthly income of residents of Daisy City.

b. To find the percentage of Daisy City's residents that have incomes higher than the mayor ($2250), we need to calculate the z-score for the mayor's income using the formula:

z = (x - μ) / σ

where x is the value we want to find the z-score for, μ is the mean, and σ is the standard deviation. In this case,

z = (2250 - 3000) / 500
= -0.75

To find the percentage of values higher than the z-score of -0.75, we can use a normal distribution table or a calculator. From the standard normal distribution table, we can find that the percentage of values higher than -0.75 is approximately 77%. Therefore, approximately 77% of Daisy City's residents have incomes that are higher than the mayor.

c. To determine the percentage of residents exempt from city taxes (incomes less than $1985 per month), we can again calculate the z-score for this income cutoff:

z = (1985 - 3000) / 500
= -2.03

Looking at the standard normal distribution table, we find that the percentage of values lower than -2.03 is approximately 0.0228, or 2.28%. Therefore, approximately 2.28% of Daisy City's residents are exempt from city taxes.

d. The minimum and maximum incomes of the middle 95% of the residents can be determined using the z-scores for the 2.5% and 97.5% percentiles.

For the 2.5% percentile (lower threshold):
z = -1.96

To find the corresponding income value, we can rearrange the z-score formula:
x = μ + (z * σ)
x = 3000 + (-1.96 * 500)
x ≈ 2052

So, the minimum income of the middle 95% of the residents is approximately $2052.

For the 97.5% percentile (upper threshold):
z = 1.96

Using the same formula:
x = μ + (z * σ)
x = 3000 + (1.96 * 500)
x ≈ 3940

So, the maximum income of the middle 95% of the residents is approximately $3940.

e. Assuming that the income distribution is representative, we can use the given information to estimate the population of Daisy City. We know that 200 residents have incomes of at least $4440 per month. We can find the z-score for this income value:

z = (4440 - 3000) / 500
= 2.88

Using the standard normal distribution table, we find that the percentage of values lower than 2.88 is approximately 0.9977. Therefore, approximately 99.77% of Daisy City's residents have incomes lower than $4440.

Let's assume that the population of Daisy City is denoted as N. Given that 99.77% of the population has incomes lower than $4440, we can set up the following equation:

0.9977 * N = 200

Solving for N:
N ≈ 200 / 0.9977
N ≈ 200.70

Therefore, the estimated population of Daisy City is approximately 200.

You have a normal distribution table handy. I do not so all I can do is outline the methods.

mean = 3000
sigma = 500
a. the monthly income
b. how far below mean is the mayor's income?
3000 - 2250 = 750
then how many sigmas below mean is this?
750/500 = 1.5 sigma (called "z") below mean
so now the question becomes how what percent of a normal distribution is between 1.5 sigma (called "z") below mean and + infinity
this is of course 100% - the percent between -oo and 1.5 sigma below mean
My guess not having the table handy is about 7% for between -oo and mean -1.5 sigma
so about 93 % have more income

c. 3000 - 1985 = 1015 below mean
1015/500 = 2.03 sigma below mean
look for between -oo and -2.03
This is tiny, around 2 1/2 percent if I had a table

d. middle 95% is from about F(z) = 2.5% to F(z)=97.5% or F(z) = .025 to .975

That would be from about mean -2 sigma to mean + 2 sigma (use table of z versus F(z) of course. Do not trust my guess)
so from mean -2*500 to mean + 2*500
or from $2000 to $4000

4440 - 3000 = 1440
1440/500 = 2.88 sigma above mean
so what percent below mean + 2.88 sigma?
F(z)about .995 (remember guessing
so
.005 or about .5% have incomes above 4440
.005 n = 200
n = 200/.005 = 40,000
remember I am beign very approximate guessing normal distribution table.