Answer :
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.
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.
