"A study has shown that 20% of all college textbooks have a price of $90 or higher. It is known that the standard deviation of the prices of all college textbooks is $9.50. Suppose the prices of all college textbooks have a normal distribution. What is the mean price of all college textbooks?"

I have S.D. = 9.5, P = 20% = .2, and I don't know where to go from here. I know the answer is $82 based on a calculator online, but I can't find out how to use the z-table to go from a probability/S.D. to a z-value to plug into X = u + S.D.*z How do I use the z-tables when given a percentage that is outside the bounds of the table? Mine only goes from .1 to .0005, so there are no listed values for .2

Take a z-chart, for example:

http://www.math.unb.ca/~knight/utility/NormTble.htm

20% of all books are $90 or higher, which means 80% of all books are $90 or lower.

Look up the chart for 0-Z (first chart in the above link) for 0.800. You'll find it on the row 0.8, and between columns 0.04 and 0.05 which gives approx. 0.842 SD from the mean after interpolation.

The mean price is therefore $90-0.842*9.5=$82.

what numbers did u plug in for z and x

in the formula z=x-u divided by the s.d

I want to this Answer

I want to explain the Answer clearly

not helpful

nvm I was mistaken, it is helpful

To find the mean price of all college textbooks, we need to use the concept of a z-score and standard normal distribution.

A z-score represents the number of standard deviations an observation or value is from the mean of a distribution. In this case, we want to find the z-score that corresponds to a percentage of 20% (or 0.2).

Since we are given that the data follows a normal distribution with a standard deviation (SD) of $9.50, we can use the standard formula to calculate the z-score:

z = (x - μ) / σ

Where:
z is the z-score
x is the observed value
μ is the mean
σ is the standard deviation

In this case, we want to find the z-score that corresponds to the 20th percentile (0.2), which represents the area under the curve up to that point. However, as you mentioned, if your z-table only goes up to a certain percentile, you can make use of the symmetry property of the standard normal distribution.

Since the normal distribution is symmetric about the mean, we can find the z-score corresponding to the 80th percentile (1 - 0.2 = 0.8) instead. This will give us the same result.

Using the z-table, find the z-score for a cumulative probability of 0.8. Locate the value closest to 0.8000 in the table, which corresponds to the z-score of around 0.84.

Now, we can solve for the mean μ using the formula:

x = μ + (z * σ)

Plug in the known values:
x = $90
σ = $9.50
z = 0.84

Rearrange the formula to solve for μ:

μ = x - (z * σ)

μ = $90 - (0.84 * $9.50)

μ ≈ $90 - $7.98

μ ≈ $82.02

Therefore, the mean price of all college textbooks is approximately $82.