The graph illustrates a normal distribution for the prices paid for a particular model of HD television. The mean price paid is $2000 and the standard deviation is $65.
What is the approximate percentage of buyers who paid between $2000 and $2065?
To find the approximate percentage of buyers who paid between $2000 and $2065, we need to calculate the z-scores corresponding to these prices and then find the area under the normal distribution curve between these z-scores.
The z-score formula is given by:
z = (x - μ) / σ
where:
x = given value ($2000 and $2065 in this case)
μ = mean price ($2000)
σ = standard deviation ($65)
For $2000:
z1 = (2000 - 2000) / 65 = 0
For $2065:
z2 = (2065 - 2000) / 65 = 1
Now, we need to find the area between z1 and z2 on the normal distribution curve. Since the distribution is symmetric, we can look up the area between 0 and 1 in the standard normal distribution table or use a calculator.
Using a standard normal distribution table or calculator, we find that the area to the left of z = 1 is approximately 0.8413, and the area to the left of z = 0 is 0.5.
To find the area between z1 and z2, we subtract the area to the left of z1 from the area to the left of z2:
Area = 0.8413 - 0.5 = 0.3413
Since the area corresponds to a percentage, we multiply by 100 to get the approximate percentage of buyers who paid between $2000 and $2065:
Approximate percentage = 0.3413 * 100 = 34.13
Therefore, approximately 34.13% of buyers paid between $2000 and $2065 for the HD television model.