A study has shown that 20.05% of all college textbooks have a price of $80 or higher. It is known that the standard deviation of the price is $10. What is the mean price of the college textbooks? Assume the price is normally distributed.
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To find the mean price of the college textbooks, you can use the concept of a normal distribution and the information provided: the percentage of textbooks with a price of $80 or higher and the standard deviation of the prices.
Here's how you can calculate it step by step:
1. Convert the percentage to a decimal: 20.05% = 0.2005.
2. Use the Z-table (also known as the standard normal distribution table) to find the corresponding Z-score for this decimal value. The Z-score represents the number of standard deviations the price of $80 or higher is from the mean.
Look up the Z-score that corresponds to a cumulative area of 0.2005 in the table. In this case, it is approximately -0.8416.
3. Now, use the Z-score formula to find the mean price (μ):
Z = (X - μ) / σ
In this formula, X is the value ($80), μ is the mean price, and σ is the standard deviation ($10).
Rearranging the formula to solve for the mean price (μ), we get:
μ = X - (Z * σ)
Plugging in the known values:
μ = $80 - (-0.8416 * $10)
Simplifying the equation:
μ = $80 + $8.416
μ = $88.416
Therefore, the mean price of the college textbooks is approximately $88.416.