the average tax charge is $95. Assuming a normal distribution as a standard deviation of $10, use the approximate areas beneath the normal curve to answer what proportion of tax prep fees were exactly $100?
To determine the proportion of tax prep fees that were exactly $100, we can use the concept of the standard normal distribution.
Given that the average tax charge is $95 and the standard deviation is $10, we need to calculate the z-score for $100. The z-score represents the number of standard deviations away from the mean.
The z-score formula is:
z = (x - μ) / σ
where:
x = value ($100)
μ = mean ($95)
σ = standard deviation ($10)
Substituting the values into the formula, we get:
z = (100 - 95) / 10
z = 0.5
The next step is to find the probability associated with the z-score of 0.5 from a standard normal distribution table or using a calculator.
Looking up the z-score of 0.5 in a standard normal distribution table, we find that the corresponding probability is 0.6915.
Therefore, approximately 0.6915 or 69.15% of tax prep fees were exactly $100.
To determine the proportion of tax prep fees that were exactly $100, we need to calculate the z-score for $100 using the given information about the average tax charge and standard deviation.
The z-score formula is:
z = (x - μ) / σ
Where:
- x represents the value we want the z-score for (in this case, $100)
- μ represents the mean (average tax charge, which is $95)
- σ represents the standard deviation of the distribution ($10)
Substituting the values into the formula:
z = (100 - 95) / 10
Now we need to find the corresponding area under the standard normal distribution curve for this z-score. Since we are looking for an exact value, we can calculate the area using the cumulative distribution function (CDF).
To find the proportion (or probability) associated with the z-score of $100, we use the z-table or a statistical software. The z-table provides the cumulative probability up to a specific z-score.
Using a standard z-table, we look up the z-score we calculated (z = (100 - 95) / 10 = 0.5) and find the corresponding area.
Upon consulting the z-table, the area to the left of z = 0.5 is approximately 0.6915.
Therefore, the proportion of tax prep fees that were exactly $100 is approximately 0.6915 or 69.15%.