The average rent in a city is $1,440 per month with a standard deviation of $150. Assume rent follows the normal distribution. [You may find it useful to reference the z table.]
a. What percentage of rents are between $1,290 and $1,590? (Round your answer to 2 decimal places.)
b. What percentage of rents are less than $1,290? (Round your answer to 2 decimal places.)
c. What percentage of rents are greater than $1,740? (Round your answer to 2 decimal places.)
To solve this problem, we need to find the z-scores corresponding to the given rent values and then determine the percentage of rents within those z-scores.
a. To find the percentage of rents between $1,290 and $1,590, we first need to find the z-scores corresponding to these values. The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the rent value, μ is the mean, and σ is the standard deviation.
For $1,290:
z = (1290 - 1440) / 150 = -1
For $1,590:
z = (1590 - 1440) / 150 = 1
Next, we need to find the percentage of rents between these z-scores. Using the z-table, we can find the percentage corresponding to z = -1 and z = 1.
From the z-table, the percentage corresponding to z = -1 is 0.1587 and the percentage corresponding to z = 1 is 0.8413.
To find the percentage of rents between these z-scores, we subtract the smaller percentage from the larger percentage:
Percentage = 0.8413 - 0.1587 = 0.6826
So, approximately 68.26% of rents are between $1,290 and $1,590.
b. To find the percentage of rents less than $1,290, we need to find the z-score for this value and look up the corresponding percentage in the z-table.
Using the same formula as before:
z = (1290 - 1440) / 150 = -1
From the z-table, the percentage corresponding to z = -1 is 0.1587.
So, approximately 15.87% of rents are less than $1,290.
c. To find the percentage of rents greater than $1,740, we need to find the z-score for this value and look up the corresponding percentage in the z-table.
Using the same formula as before:
z = (1740 - 1440) / 150 = 2
From the z-table, the percentage corresponding to z = 2 is 0.9772.
So, approximately 97.72% of rents are greater than $1,740.
To solve these questions, we can use the properties of the normal distribution and the standard z-score formula.
a. To find the percentage of rents between $1,290 and $1,590, we need to calculate the z-scores for these values and use the z-table to find the corresponding percentages.
The z-score formula is:
z = (x - μ) / σ
where:
x is the value of interest ($1,290 or $1,590),
μ is the mean ($1,440),
σ is the standard deviation ($150).
For $1,290:
z1 = ($1,290 - $1,440) / $150
For $1,590:
z2 = ($1,590 - $1,440) / $150
Now, we can consult the z-table to find the corresponding percentages for z1 and z2. The z-table provides the area under the standard normal curve to the left of a given z-score.
Let's calculate z1 and z2:
z1 = ($1,290 - $1,440) / $150
z1 = -1
z2 = ($1,590 - $1,440) / $150
z2 = 1
Using the z-table, we can find the percentage of rents between these two values by subtracting the cumulative percentage below z1 from the cumulative percentage below z2:
Percentage = Cumulative % at z2 - Cumulative % at z1
Using the z-table, the cumulative percentage below z1 is 0.1587, and the cumulative percentage below z2 is 0.8413.
Therefore, the percentage of rents between $1,290 and $1,590 is:
Percentage = 0.8413 - 0.1587
Percentage = 0.6826
So, approximately 68.26% of rents are between $1,290 and $1,590.
b. To find the percentage of rents less than $1,290, we can calculate the z-score using the same formula as in part a:
z = ($1,290 - $1,440) / $150
z = -1
Using the z-table, we can find the cumulative percentage below z:
Percentage = Cumulative % at z
The cumulative percentage below z is 0.1587.
Therefore, the percentage of rents less than $1,290 is approximately 15.87%.
c. To find the percentage of rents greater than $1,740, we can calculate the z-score using the same formula as in part a:
z = ($1,740 - $1,440) / $150
z = 2
Using the z-table, we can find the cumulative percentage below z and subtract it from 1 to find the percentage above z:
Percentage = 1 - Cumulative % at z
The cumulative percentage below z is 0.9772.
Therefore, the percentage of rents greater than $1,740 is approximately 1 - 0.9772 = 0.0228, or approximately 2.28%.
In summary:
a. Approximately 68.26% of rents are between $1,290 and $1,590.
b. Approximately 15.87% of rents are less than $1,290.
c. Approximately 2.28% of rents are greater than $1,740.
To answer these questions, we will need to use the z-score formula and the standard normal distribution table (also known as the z-table). The z-score formula is given by:
z = (x - μ) / σ
where:
- x is the value we are interested in,
- μ is the population mean (in this case, the average rent),
- σ is the population standard deviation.
Now, let's calculate the z-scores for the given values.
a. To find the percentage of rents between $1,290 and $1,590, we need to find the area under the curve between these two values. First, we calculate the z-scores for both values:
z1 = (1290 - 1440) / 150 = -1
z2 = (1590 - 1440) / 150 = 1
The z-table provides the probabilities associated with z-scores. From the table, we find that the area to the left of z1 is 0.1587 and the area to the left of z2 is 0.8413. As the normal distribution is symmetric, the area between z1 and z2 is:
area = 0.8413 - 0.1587 = 0.6826
So, approximately 68.26% of rents are between $1,290 and $1,590.
b. To find the percentage of rents less than $1,290, we need to find the area under the curve to the left of this value. Calculating the corresponding z-score:
z = (1290 - 1440) / 150 = -1
From the z-table, we find that the area to the left of z is 0.1587. Therefore, approximately 15.87% of rents are less than $1,290.
c. To find the percentage of rents greater than $1,740, we need to find the area under the curve to the right of this value. Calculating the corresponding z-score:
z = (1740 - 1440) / 150 = 2
From the z-table, we find that the area to the left of z is 0.9772. However, we want the area to the right, so we subtract this value from 1:
area = 1 - 0.9772 = 0.0228
So, approximately 2.28% of rents are greater than $1,740.
Remember to round your answers to 2 decimal places as stated in the question.