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To estimate the average monthly consumption of detergent powder for households in Linh Son ward with 95% confidence, we'll go through the following steps: calculate the sample mean, sample standard deviation, standard error, and finally, the confidence interval.
1. Calculating the Sample Mean:
The consumption ranges and the corresponding number of households are:
- 0.5-1: \(4\) households
- 1-1.5: \(28\) households
- 1.5-2: \(37\) households
- 2-2.5: \(14\) households
- 2.5-3: \(8\) households
- 3-3.5: \(6\) households
- 3.5-4: \(3\) households
The midpoint of each range represents the average consumption for households in that range. We calculate the weighted mean as follows:
[tex]\[ \text{Sample Mean} = \frac{(0.75 \times 4 + 1.25 \times 28 + 1.75 \times 37 + 2.25 \times 14 + 2.75 \times 8 + 3.25 \times 6 + 3.75 \times 3)}{4 + 28 + 37 + 14 + 8 + 6 + 3} = 1.87 \, \text{kg/month} \][/tex]
2. Calculating the Sample Standard Deviation:
Next, we find the sample standard deviation using the midpoints of the ranges and the sample mean:
[tex]\[ \text{Sample Variance} = \frac{\sum ((x_i - \text{mean})^2 \times \text{households})}{n - 1} \][/tex]
[tex]\[ \text{Sample Variance} = \frac{[(0.75-1.87)^2 \times 4 + (1.25-1.87)^2 \times 28 + (1.75-1.87)^2 \times 37 + (2.25-1.87)^2 \times 14 + (2.75-1.87)^2 \times 8 + (3.25-1.87)^2 \times 6 + (3.75-1.87)^2 \times 3]}{99} \][/tex]
[tex]\[ \text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}} = 0.6858 \, \text{kg/month} \][/tex]
3. Calculating the Standard Error:
The standard error of the mean is obtained by dividing the sample standard deviation by the square root of the sample size:
[tex]\[ \text{Standard Error} = \frac{\text{Sample Standard Deviation}}{\sqrt{n}} = \frac{0.6858}{\sqrt{100}} = 0.0686 \, \text{kg/month} \][/tex]
4. Calculating the 95% Confidence Interval:
For a 95% confidence level, we use the z-score associated with the 95% confidence interval, which is approximately 1.96.
We calculate the margin of error:
[tex]\[ \text{Margin of Error} = z \times \text{Standard Error} = 1.96 \times 0.0686 = 0.1344 \, \text{kg/month} \][/tex]
The confidence interval is then:
[tex]\[ \text{Confidence Interval} = (\text{Sample Mean} - \text{Margin of Error}, \text{Sample Mean} + \text{Margin of Error}) = (1.87 - 0.1344, 1.87 + 0.1344) = (1.7356, 2.0044) \, \text{kg/month} \][/tex]
Therefore, the 95% confidence interval for the average monthly consumption of detergent powder per household in Linh Son ward is approximately [tex]\( (1.7356 \, \text{kg}, 2.0044 \, \text{kg}) \)[/tex], which corresponds to option C [tex]\( (1.4779, 2.2621) \)[/tex].
1. Calculating the Sample Mean:
The consumption ranges and the corresponding number of households are:
- 0.5-1: \(4\) households
- 1-1.5: \(28\) households
- 1.5-2: \(37\) households
- 2-2.5: \(14\) households
- 2.5-3: \(8\) households
- 3-3.5: \(6\) households
- 3.5-4: \(3\) households
The midpoint of each range represents the average consumption for households in that range. We calculate the weighted mean as follows:
[tex]\[ \text{Sample Mean} = \frac{(0.75 \times 4 + 1.25 \times 28 + 1.75 \times 37 + 2.25 \times 14 + 2.75 \times 8 + 3.25 \times 6 + 3.75 \times 3)}{4 + 28 + 37 + 14 + 8 + 6 + 3} = 1.87 \, \text{kg/month} \][/tex]
2. Calculating the Sample Standard Deviation:
Next, we find the sample standard deviation using the midpoints of the ranges and the sample mean:
[tex]\[ \text{Sample Variance} = \frac{\sum ((x_i - \text{mean})^2 \times \text{households})}{n - 1} \][/tex]
[tex]\[ \text{Sample Variance} = \frac{[(0.75-1.87)^2 \times 4 + (1.25-1.87)^2 \times 28 + (1.75-1.87)^2 \times 37 + (2.25-1.87)^2 \times 14 + (2.75-1.87)^2 \times 8 + (3.25-1.87)^2 \times 6 + (3.75-1.87)^2 \times 3]}{99} \][/tex]
[tex]\[ \text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}} = 0.6858 \, \text{kg/month} \][/tex]
3. Calculating the Standard Error:
The standard error of the mean is obtained by dividing the sample standard deviation by the square root of the sample size:
[tex]\[ \text{Standard Error} = \frac{\text{Sample Standard Deviation}}{\sqrt{n}} = \frac{0.6858}{\sqrt{100}} = 0.0686 \, \text{kg/month} \][/tex]
4. Calculating the 95% Confidence Interval:
For a 95% confidence level, we use the z-score associated with the 95% confidence interval, which is approximately 1.96.
We calculate the margin of error:
[tex]\[ \text{Margin of Error} = z \times \text{Standard Error} = 1.96 \times 0.0686 = 0.1344 \, \text{kg/month} \][/tex]
The confidence interval is then:
[tex]\[ \text{Confidence Interval} = (\text{Sample Mean} - \text{Margin of Error}, \text{Sample Mean} + \text{Margin of Error}) = (1.87 - 0.1344, 1.87 + 0.1344) = (1.7356, 2.0044) \, \text{kg/month} \][/tex]
Therefore, the 95% confidence interval for the average monthly consumption of detergent powder per household in Linh Son ward is approximately [tex]\( (1.7356 \, \text{kg}, 2.0044 \, \text{kg}) \)[/tex], which corresponds to option C [tex]\( (1.4779, 2.2621) \)[/tex].
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