The amount of time a certain brand of light bulb lasts is normally distributed with a mean of 1100 hours and a standard deviation of 60 hours. Out of 375 freshly installed light bulbs in a new large building, how many would be expected to last between 1050 hours and 1140 hours, to the nearest whole number?

To solve this problem, we will use the properties of the normal distribution. The problem specifies that the life of the light bulbs is normally distributed with a mean of 1100 hours and a standard deviation of 60 hours. We're asked to find how many bulbs out of 375 would last between 1050 hours and 1140 hours.

### Step 1: Standardize the values
We'll first convert the hours (1050 and 1140) into z-scores using the formula:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \( x \) is the value we are standardizing,
- \( \mu = 1100 \) is the mean,
- \( \sigma = 60 \) is the standard deviation.

#### For \( x = 1050 \):
\[
z_1 = \frac{1050 - 1100}{60} = \frac{-50}{60} = -0.8333
\]

#### For \( x = 1140 \):
\[
z_2 = \frac{1140 - 1100}{60} = \frac{40}{60} = 0.6667
\]

### Step 2: Find the cumulative probabilities
Next, we'll find the cumulative probabilities corresponding to these z-scores using a standard normal distribution table or a calculator.

- For \( z_1 = -0.8333 \), the cumulative probability \( P(z_1) \) is approximately 0.2023.
- For \( z_2 = 0.6667 \), the cumulative probability \( P(z_2) \) is approximately 0.7475.

### Step 3: Calculate the probability between the two z-scores
The probability that a bulb lasts between 1050 and 1140 hours is the difference between the two cumulative probabilities:

\[
P(1050 \leq X \leq 1140) = P(z_2) - P(z_1) = 0.7475 - 0.2023 = 0.5452
\]

This means that 54.52% of the light bulbs are expected to last between 1050 and 1140 hours.

### Step 4: Calculate the expected number of bulbs
Now, we find how many bulbs out of the 375 would fall in this range:

\[
\text{Expected number of bulbs} = 0.5452 \times 375 = 204.45
\]

Rounding to the nearest whole number, we expect **204** light bulbs to last between 1050 and 1140 hours.

### Conclusion
The expected number of light bulbs to last between 1050 hours and 1140 hours is 204.

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Would you like more details on this process or have any other questions?

Here are 5 related questions to deepen your understanding:
1. What is the probability of a bulb lasting more than 1140 hours?
2. How many bulbs would last less than 1000 hours?
3. What is the z-score for a bulb that lasts 1200 hours?
4. How does the standard deviation affect the spread of the distribution?
5. What is the total probability for all bulbs lasting between 950 and 1250 hours?

**Tip:** In normal distributions, about 68% of the data lies within one standard deviation of the mean. This fact helps approximate probabilities without calculations.

Discover how to calculate the expected number of light bulbs that last between 1050 and 1140 hours, using a normal distribution with a mean of 1100 hours and standard deviation of 60 hours. This solution involves standardizing values and applying probability theory to find the result. Suitable for high school students learning statistics.

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