The daily demand for books at House Bookstore is a normal random variable with a mean of 200 books. On 34% of the days they are open, they sell between 200 and 215 books. On what percentage of the days they are open do they sell fewer than 185 books?

To solve this problem, we'll use the properties of the normal distribution. Let's break down the steps:

### Step 1: Define the problem
Given:
- The daily demand for books is a normal random variable \( X \) with a mean \( \mu = 200 \) books.
- The standard deviation \( \sigma \) is unknown.
- On 34% of the days, the demand is between 200 and 215 books.

We need to find the percentage of days when the demand is fewer than 185 books.

### Step 2: Determine the standard deviation
To find the standard deviation \( \sigma \), we use the information given for the 34% probability:

1. Standardize the values 200 and 215 using the z-score formula:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   For \( X = 200 \), \( Z_1 = \frac{200 - 200}{\sigma} = 0 \).
   
   For \( X = 215 \), \( Z_2 = \frac{215 - 200}{\sigma} = \frac{15}{\sigma} \).

2. We know that the area between these z-scores (0 and \( Z_2 \)) is 34%. Looking up the z-value corresponding to 34% in the positive tail of the normal distribution, we get \( Z_2 \approx 0.415 \) (because the total area from the mean to this point must be 0.5 + 0.34 = 0.84).

   Therefore:
   \[
   \frac{15}{\sigma} = 0.415
   \]
   Solving for \( \sigma \):
   \[
   \sigma = \frac{15}{0.415} \approx 36.14
   \]

### Step 3: Calculate the z-score for 185 books
Next, we calculate the z-score for 185 books:
\[
Z = \frac{185 - 200}{36.14} \approx \frac{-15}{36.14} \approx -0.415
\]

### Step 4: Find the percentage of days with fewer than 185 books
Now, we find the cumulative probability associated with \( Z = -0.415 \). Using the standard normal distribution table or a calculator:

\[
P(Z < -0.415) \approx 0.339
\]

This corresponds to approximately 33.9% of the days.

### Conclusion
On approximately **33.9%** of the days, House Bookstore sells fewer than 185 books.

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Do you want details or have any questions? 

Here are 5 related questions you might find interesting:
1. How would the percentage change if the mean demand increased to 220 books?
2. What is the probability that the bookstore sells between 185 and 215 books in a day?
3. How would the standard deviation change if the range for 34% of days shifted to between 190 and 205 books?
4. What if the bookstore wanted to know the probability of selling more than 215 books on any given day?
5. How can this method be used to calculate demand over multiple days rather than a single day?

**Tip:** Understanding the properties of the normal distribution, like the empirical rule (68-95-99.7), can help you quickly estimate probabilities and percentages.

Calculate the probability that House Bookstore sells fewer than 185 books based on a normal distribution with a mean of 200 books. Understand the application of Z-scores and cumulative probability in solving this problem.

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