To price her books to sell to the middle 80% range, the publisher needs to set the minimum and maximum prices such that 10% of the prices fall below the minimum price and 10% of the prices exceed the maximum price. This leaves 80% of the prices in the middle range.
To find the minimum and maximum prices, we need to use the z-score formula:
z = (x - μ) / σ
where z is the z-score, x is the price, μ is the mean price, and σ is the standard deviation.
To find the z-score corresponding to the 10th and 90th percentiles, we consult a standard normal distribution table and find that:
z_10 = -1.28
z_90 = 1.28
Substituting the given values, we get:
-1.28 = (x - 12.80) / 0.83
1.28 = (x - 12.80) / 0.83
Solving for x, we get:
x = -1.28(0.83) + 12.80 = $11.72 (rounded to the nearest cent) for the minimum price
x = 1.28(0.83) + 12.80 = $13.88 (rounded to the nearest cent) for the maximum price
Therefore, the minimum price should be $11.72 and the maximum price should be $13.88 to sell to the middle 80% range.