suppose that prices of a certain model of new homes are normally distributed with a mean of $150,000 use 68-95-99.7 rule to find the percentage of buyers who paid:
between $150,000 and $152,000 if the standard deviation is $1200
To find the percentage of buyers who paid between $150,000 and $152,000, we need to calculate the z-scores for both prices and use the z-table to determine the corresponding percentages.
Step 1: Calculate the z-score for the lower price of $150,000:
z = (x - mean) / standard deviation
z = ($150,000 - $150,000) / $1,200
z = 0
Step 2: Calculate the z-score for the higher price of $152,000:
z = (x - mean) / standard deviation
z = ($152,000 - $150,000) / $1,200
z = 1.67
Step 3: Use the z-table to find the corresponding percentages for the z-scores from Step 1 and Step 2.
a) For the lower price z-score of 0:
Using the z-table, the percentage corresponding to a z-score of 0 is 0.5000, or 50%.
b) For the higher price z-score of 1.67:
Using the z-table, the percentage corresponding to a z-score of 1.67 is approximately 0.9525, or 95.25%.
Step 4: Calculate the percentage of buyers who paid between $150,000 and $152,000:
To find the percentage between these two prices, subtract the percentage corresponding to the lower price from the percentage corresponding to the higher price:
Percentage = Percentage(higher price) - Percentage(lower price)
Percentage = 95.25% - 50%
Percentage = 45.25%
Therefore, approximately 45.25% of the buyers paid between $150,000 and $152,000 for the new homes.
To find the percentage of buyers who paid between $150,000 and $152,000, we need to calculate the z-scores for both prices and then use the 68-95-99.7 rule.
Step 1: Calculate the z-score for $150,000.
The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
In this case, x = $150,000, μ = $150,000, and σ = $1,200.
Plugging these values into the formula, we get:
z = ($150,000 - $150,000) / $1,200
z = 0
Step 2: Calculate the z-score for $152,000.
Using the same formula as above, we get:
z = ($152,000 - $150,000) / $1,200
z = $2,000 / $1,200
z = 1.67
Step 3: Use the 68-95-99.7 rule to find the percentage of buyers in the range between $150,000 and $152,000.
According to the 68-95-99.7 rule:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% falls within 2 standard deviations.
- Approximately 99.7% falls within 3 standard deviations.
Since both $150,000 and $152,000 are within 1 standard deviation of the mean ($150,000), we can conclude that roughly 68% of the buyers paid between $150,000 and $152,000.
you can plug in your numbers and play around with Z table stuff at
http://davidmlane.com/hyperstat/z_table.html