suppose that prices of a certain model of new homes are normally distributed with a mean of $150,000 use the rule to find the percentage of buyers who paid:
between $150,000if the standard deviation is $1200
It would help if you proofread your questions before you posted them.
"Between $150,000" and what?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Multiply by 100.
To find the percentage of buyers who paid between $150,000, we need to calculate the z-scores for the given values.
The z-score is calculated using the formula: z = (x - μ) / σ
where:
x = the given value
μ = the mean
σ = the standard deviation
Z-score for $150,000:
z = (150,000 - 150,000) / 1,200
Since the value is equal to the mean, the z-score would be 0.
To find the percentage of buyers who paid between $150,000, we can refer to the standard normal distribution table. In this case, we need to find the area under the curve between a z-score of 0 and the right tail.
From the standard normal distribution table, the area to the left of a z-score of 0 is 0.5000. Therefore, the percentage of buyers who paid between $150,000 is:
Percentage = 0.5000 * 100%
Percentage = 50%
So, approximately 50% of the buyers paid between $150,000 for the certain model of new homes.
To find the percentage of buyers who paid between two specific prices, we can use the empirical rule, also known as the 68-95-99.7 rule, which applies to normal distributions.
The empirical rule states that:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In this case, the mean price of the new homes is $150,000, and the standard deviation is $1200.
To find the percentage of buyers who paid between $150,000, we need to consider one standard deviation above and below the mean.
One standard deviation above the mean:
Mean + (1 * Standard Deviation) = $150,000 + ($1200 * 1) = $150,000 + $1200 = $151,200
One standard deviation below the mean:
Mean - (1 * Standard Deviation) = $150,000 - ($1200 * 1) = $150,000 - $1200 = $148,800
So, the range of prices between $148,800 and $151,200 represents one standard deviation from the mean.
Using the empirical rule, we can say that approximately 68% of the buyers paid between these two prices.