Answer :
To estimate the average demand for detergent consumption among households in Lĩnh Sơn ward with 95% confidence, let's follow these steps:
1. Midpoint Calculation:
First, we need to find the midpoint of each interval for the demand:
- For the interval \(0.5-1\), the midpoint is \( (0.5 + 1) / 2 = 0.75 \).
- For the interval \(1-1.5\), the midpoint is \( (1 + 1.5) / 2 = 1.25 \).
- For the interval \(1.5-2\), the midpoint is \( (1.5 + 2) / 2 = 1.75 \).
- For the interval \(2-2.5\), the midpoint is \( (2 + 2.5) / 2 = 2.25 \).
- For the interval \(2.5-3\), the midpoint is \( (2.5 + 3) / 2 = 2.75 \).
- For the interval \(3-3.5\), the midpoint is \( (3 + 3.5) / 2 = 3.25 \).
- For the interval \(3.5-4\), the midpoint is \( (3.5 + 4) / 2 = 3.75 \).
2. Given Data:
- Midpoints: \([0.75, 1.25, 1.75, 2.25, 2.75, 3.25, 3.75]\)
- Number of Households (Frequencies): \([4, 28, 37, 14, 8, 6, 3]\)
- Total Number of Households in the ward: 1000
3. Sample Mean Calculation:
The sample mean \(\bar{x}\) is calculated using the formula:
[tex]\[ \bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \][/tex]
where \(x_i\) is the midpoint and \(f_i\) is the frequency. The calculation gives us:
[tex]\[ \bar{x} = 1.87 \][/tex]
4. Sample Variance Calculation:
The sample variance \(s^2\) is calculated using the formula:
[tex]\[ s^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i - 1} \][/tex]
This results in:
[tex]\[ s^2 = 0.4703 \][/tex]
5. Sample Standard Deviation Calculation:
The sample standard deviation \(s\) is the square root of the sample variance:
[tex]\[ s = \sqrt{s^2} = 0.6858 \][/tex]
6. Standard Error of the Mean:
The standard error of the mean (SE) is calculated as:
[tex]\[ SE = \frac{s}{\sqrt{n}} \][/tex]
where \(n\) is the number of data points (total number of households surveyed, not the total in the ward):
[tex]\[ SE = 0.0686 \][/tex]
7. 95% Confidence Interval Calculation:
The z-value for a 95% confidence interval is 1.96. The confidence interval for the mean is calculated as:
[tex]\[ \text{Lower Bound} = \bar{x} - (z \cdot SE) = 1.7356 \][/tex]
[tex]\[ \text{Upper Bound} = \bar{x} + (z \cdot SE) = 2.0044 \][/tex]
So, with 95% confidence, the average monthly detergent consumption per household in Lĩnh Sơn ward is estimated to be between 1.7356 kg and 2.0044 kg.
1. Midpoint Calculation:
First, we need to find the midpoint of each interval for the demand:
- For the interval \(0.5-1\), the midpoint is \( (0.5 + 1) / 2 = 0.75 \).
- For the interval \(1-1.5\), the midpoint is \( (1 + 1.5) / 2 = 1.25 \).
- For the interval \(1.5-2\), the midpoint is \( (1.5 + 2) / 2 = 1.75 \).
- For the interval \(2-2.5\), the midpoint is \( (2 + 2.5) / 2 = 2.25 \).
- For the interval \(2.5-3\), the midpoint is \( (2.5 + 3) / 2 = 2.75 \).
- For the interval \(3-3.5\), the midpoint is \( (3 + 3.5) / 2 = 3.25 \).
- For the interval \(3.5-4\), the midpoint is \( (3.5 + 4) / 2 = 3.75 \).
2. Given Data:
- Midpoints: \([0.75, 1.25, 1.75, 2.25, 2.75, 3.25, 3.75]\)
- Number of Households (Frequencies): \([4, 28, 37, 14, 8, 6, 3]\)
- Total Number of Households in the ward: 1000
3. Sample Mean Calculation:
The sample mean \(\bar{x}\) is calculated using the formula:
[tex]\[ \bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \][/tex]
where \(x_i\) is the midpoint and \(f_i\) is the frequency. The calculation gives us:
[tex]\[ \bar{x} = 1.87 \][/tex]
4. Sample Variance Calculation:
The sample variance \(s^2\) is calculated using the formula:
[tex]\[ s^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i - 1} \][/tex]
This results in:
[tex]\[ s^2 = 0.4703 \][/tex]
5. Sample Standard Deviation Calculation:
The sample standard deviation \(s\) is the square root of the sample variance:
[tex]\[ s = \sqrt{s^2} = 0.6858 \][/tex]
6. Standard Error of the Mean:
The standard error of the mean (SE) is calculated as:
[tex]\[ SE = \frac{s}{\sqrt{n}} \][/tex]
where \(n\) is the number of data points (total number of households surveyed, not the total in the ward):
[tex]\[ SE = 0.0686 \][/tex]
7. 95% Confidence Interval Calculation:
The z-value for a 95% confidence interval is 1.96. The confidence interval for the mean is calculated as:
[tex]\[ \text{Lower Bound} = \bar{x} - (z \cdot SE) = 1.7356 \][/tex]
[tex]\[ \text{Upper Bound} = \bar{x} + (z \cdot SE) = 2.0044 \][/tex]
So, with 95% confidence, the average monthly detergent consumption per household in Lĩnh Sơn ward is estimated to be between 1.7356 kg and 2.0044 kg.
