Questions: A company wants to price its item in the top 2% price range. If the mean price of the items is 126 with a standard deviation of 13, what is the lowest price the company can charge and still be in the top 2%?

A company wants to price its item in the top 2% price range. If the mean price of the items is 126 with a standard deviation of 13, what is the lowest price the company can charge and still be in the top 2%?

Transcript text: A company wants to price its item in the top $2 \%$ price range. If the mean price of the items is $\$ 126$ with a standard deviation of $\$ 13$, what is the lowest price the company can charge and still be in the top $2 \%$ ?

Solution

Solution Steps

To find the lowest price the company can charge to be in the top 2% of prices, we need to determine the z-score that corresponds to the 98th percentile of a standard normal distribution. Then, we use this z-score to calculate the price using the given mean and standard deviation.

Step 1: Determine the Z-Score

To find the lowest price in the top 2% of prices, we first need to find the z-score corresponding to the 98th percentile of a standard normal distribution. This z-score is calculated as: \[ z = 2.0537 \]

Step 2: Calculate the Lowest Price

Using the z-score, we can calculate the lowest price $ P $ that corresponds to this z-score using the formula: \[ P = \mu + z \cdot \sigma \] where $ \mu = 126 $ (mean price) and $ \sigma = 13 $ (standard deviation). Substituting the values, we have: \[ P = 126 + 2.0537 \cdot 13 \] Calculating this gives: \[ P \approx 126 + 26.6981 \approx 152.6987 \]