A purchasing manager at a university is investigating which brand of LCD projector to purchase for equipping classrooms. Of major concern to her is the lifetime of the light bulbs used in the projectors. One company has published the following frequency distribution from a test of the lifetimes (in hours) of 43 bulbs used in its LCD projectors.

\begin{tabular}{|c|c|}
\hline
\begin{tabular}{l}
Bulb lifetime \\
(in hours)
\end{tabular} & Frequency \\
\hline 700 to 749 & 5 \\
\hline 750 to 799 & 8 \\
\hline 800 to 849 & 12 \\
\hline 850 to 899 & 9 \\
\hline 900 to 949 & 5 \\
\hline 950 to 999 & 4 \\
\hline
\end{tabular}

Based on the frequency distribution, using the midpoint of each data class, estimate the mean lifetime for the light bulbs in the company's test. For your intermediate computations, use four or more decimal places, and round your answer to one decimal place.

[tex]$\square$[/tex] hours

To estimate the mean lifetime of the light bulbs based on the given frequency distribution, we will follow these steps:

1. Identify the midpoints for each class interval: The midpoint of each class interval [tex]$(a, b)$[/tex] is calculated using the formula:
[tex]\[ \text{midpoint} = \frac{a + b}{2} \][/tex]

2. Multiply each midpoint by its corresponding frequency: This helps to calculate the weighted contribution of each class interval to the overall mean.

3. Calculate the total frequency: This is the sum of all frequencies.

4. Calculate the sum of the products of midpoints and frequencies: This is done by summing up the values obtained from multiplying the midpoint by the frequency for each class interval.

5. Compute the mean: The mean lifetime is the ratio of the sum calculated in step 4 to the total frequency from step 3. The formula for the mean is:
[tex]\[ \text{mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\text{total frequency}} \][/tex]

Now, let's apply these steps to the given data:

### Step 1: Calculate Midpoints
- For the interval 700 to 749:
[tex]\[ \text{midpoint} = \frac{700 + 749}{2} = 724.5 \][/tex]
- For the interval 750 to 799:
[tex]\[ \text{midpoint} = \frac{750 + 799}{2} = 774.5 \][/tex]
- For the interval 800 to 849:
[tex]\[ \text{midpoint} = \frac{800 + 849}{2} = 824.5 \][/tex]
- For the interval 850 to 899:
[tex]\[ \text{midpoint} = \frac{850 + 899}{2} = 874.5 \][/tex]
- For the interval 900 to 949:
[tex]\[ \text{midpoint} = \frac{900 + 949}{2} = 924.5 \][/tex]
- For the interval 950 to 999:
[tex]\[ \text{midpoint} = \frac{950 + 999}{2} = 974.5 \][/tex]

### Step 2: Multiply Midpoints by Corresponding Frequencies
- For [tex]$724.5 \times 5 = 3622.5$[/tex]
- For [tex]$774.5 \times 8 = 6196.0$[/tex]
- For [tex]$824.5 \times 12 = 9894.0$[/tex]
- For [tex]$874.5 \times 9 = 7870.5$[/tex]
- For [tex]$924.5 \times 5 = 4622.5$[/tex]
- For [tex]$974.5 \times 4 = 3898.0$[/tex]

### Step 3: Calculate the Total Frequency
[tex]\[ \text{Total frequency} = 5 + 8 + 12 + 9 + 5 + 4 = 43 \][/tex]

### Step 4: Calculate the Sum of the Products of Midpoints and Frequencies
[tex]\[ \text{Sum of products} = 3622.5 + 6196.0 + 9894.0 + 7870.5 + 4622.5 + 3898.0 = 36103.5 \][/tex]

### Step 5: Calculate the Mean
[tex]\[ \text{mean} = \frac{36103.5}{43} \approx 839.6 \][/tex]

### Conclusion
The estimated mean lifetime for the light bulbs in the company's test is [tex]$839.6$[/tex] hours.