Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.