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.

I am using z= x-the mean/sd but i am not getting a correct answer.

I find this page very useful for these kind of problems.

http://davidmlane.com/hyperstat/z_table.html

The beauty is that you don't even have to enter the z-scores (you can of course), but rather enter the mean and SD directly.

Since these kind of question rely on the use of tables or charts, I don't see any difference in using tables or a new type of application as this applet.
Let me know how it works for you

BTW, bobpursley answered the post just below this one for you yesterday.

i used the site (thx a bunch) but after i get the area i am confused as to how i get the final answer? the area for b is .066807 but i am not sure what to do next

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

b. To calculate the percentage of Daisy City's residents who have incomes that are more than the mayor ($2250), we need to find the z-score for this value.

First, we calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.

In this case, x = $2250, μ = $3000, and σ = $500.

z = ($2250 - $3000) / $500
z = -0.15

Next, we need to find the area to the right of the z-score (-0.15) on the standard normal distribution table or using a calculator. The area to the right of the z-score is the percentage of residents with incomes higher than the mayor.

Using a standard normal table, we find that the area to the right of -0.15 is 0.5596, or 55.96%. Therefore, approximately 55.96% of Daisy City's residents have incomes that are more than the mayor.

c. To calculate the percentage of residents exempt from city taxes (with incomes less than $1985), we need to find the z-score for this value.

Using the formula z = (x - μ) / σ, where x = $1985, μ = $3000, and σ = $500:

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

Next, we find the area to the left of the z-score (-2.03) on the standard normal distribution table or using a calculator. The area to the left of the z-score is the percentage of residents exempt from city taxes.

Using a standard normal table, we find that the area to the left of -2.03 is 0.0212, or 2.12%. Therefore, approximately 2.12% of the residents in Daisy City are exempt from city taxes.

d. The middle 95% of the residents refers to the range of incomes that fall within two standard deviations from the mean.

To calculate the minimum and maximum incomes of the middle 95%, we need to find the z-scores corresponding to the cumulative probability of 0.025 (for the lower end) and 0.975 (for the upper end) on the standard normal distribution table or using a calculator.

Using a standard normal table, the z-scores for the cumulative probabilities of 0.025 and 0.975 are approximately -1.96 and 1.96, respectively.

Minimum income:
z = -1.96
x = z * σ + μ
x = -1.96 * $500 + $3000
x ≈ $1980.20 (rounded to $1980)

Maximum income:
z = 1.96
x = z * σ + μ
x = 1.96 * $500 + $3000
x ≈ $4019.80 (rounded to $4020)

Therefore, the minimum and maximum incomes for the middle 95% of Daisy City residents are approximately $1980 and $4020, respectively.

e. To determine the population of Daisy City given that there are 200 residents with incomes of at least $4440 per month, we can use the provided mean and standard deviation.

Using the z-score formula: z = (x - μ) / σ

For x = $4440, μ = $3000, and σ = $500:
z = ($4440 - $3000) / $500
z = 2.88

The z-score of 2.88 indicates that the income of the residents that have at least $4440 per month is approximately 2.88 standard deviations above the mean.

Using a standard normal table or a calculator, the area to the right of 2.88 is 0.0020, or 0.20%.

Since we know that there are 200 residents in the city meeting this criterion, we can calculate the population of Daisy City using the formula: population = (number of residents / percentage).

Therefore, the population of Daisy City is approximately 200 / 0.0020 = 100,000 residents.

To answer these questions, you can use the Z-score formula, as you mentioned:

Z = (X - mean) / standard deviation

Now let's go through each question step by step:

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

b. To find the percentage of residents with incomes that are more than the mayor's income ($2250), we need to calculate the Z-score for this value and find the percentage of values higher than it.

Z = ($2250 - $3000) / $500
Z = -1.5

Using a Z-table or a calculator, you can find the percentage associated with a Z-score of -1.5, which is approximately 6.68%. Therefore, about 6.68% of Daisy City's residents have incomes higher than the mayor.

c. To find the percentage of residents exempt from city taxes (incomes less than $1985), we need to calculate the Z-score for this value and find the percentage of values lower than it.

Z = ($1985 - $3000) / $500
Z = -2.03

Using a Z-table or a calculator, you can find the percentage associated with a Z-score of -2.03, which is approximately 1.97%. Therefore, about 1.97% of Daisy City's residents are exempt from city taxes.

d. The middle 95% of the population falls within 2 standard deviations from the mean. In other words, we need to find the values that correspond to the 2.5th percentile and the 97.5th percentile. These values will represent the minimum and maximum incomes, respectively, of the middle 95% of residents.

First, find the Z-scores for these percentiles:

For the 2.5th percentile: Z = -1.96
For the 97.5th percentile: Z = 1.96

Now, use the Z-score formula to calculate the corresponding values:

Minimum income = (Z * standard deviation) + mean
Minimum income = (-1.96 * $500) + $3000
Minimum income ≈ $1002

Maximum income = (Z * standard deviation) + mean
Maximum income = (1.96 * $500) + $3000
Maximum income ≈ $4998

Therefore, the minimum and maximum incomes of the middle 95% of the residents are approximately $1002 and $4998, respectively.

e. To find the population of Daisy City, we can estimate it based on the given information. Let's assume the distribution of incomes in Daisy City follows the normal distribution. Since we know the mean and standard deviation, we can use the Z-score formula to calculate the percentage of residents with incomes of at least $4440.

Z = ($4440 - $3000) / $500
Z = 2.88

Using a Z-table or a calculator, you can find the percentage associated with a Z-score of 2.88, which is approximately 99.92%. Therefore, approximately 99.92% of Daisy City's residents have incomes of at least $4440.

Now let's calculate the population using this percentage:

x residents have an income of at least $4440
0.9992x = 200 (we want to find 'x')

Solving the equation:
x ≈ 200 / 0.9992
x ≈ 200,240

Therefore, the estimated population of Daisy City is approximately 200,240 based on the assumption that the distribution of incomes follows a normal distribution.