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.
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.)
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