A pollution-control inspector suspected that a riverside community was releasing semi treated sewage into a river and this, as a consequence, was changing the level of dissolved oxygen of the river. To check this, he drew 15 randomly selected specimens of river water at a location above the town, and another 15 specimens the town. For the specimens above the town the mean dissolved oxygen reading was 4.92 with a standard deviation of 0.157. For the specimens below the town, the mean dissolved oxygen reading was 4.74 with a standard deviation of 0.32. The upper limit of a 95% confidence interval for the mean differences of the two populations is
To find the upper limit of a 95% confidence interval for the mean difference of the two populations, we can use the formula:
Upper Limit = (Mean Difference) + (Z-Score) * (Standard Error)
First, let's calculate the mean difference:
Mean Difference = Mean of samples above the town - Mean of samples below the town
= 4.92 - 4.74
= 0.18
Next, let's calculate the standard error:
Standard Error = sqrt((Standard Deviation above the town)^2/n1 + (Standard Deviation below the town)^2/n2)
= sqrt((0.157)^2/15 + (0.32)^2/15)
= sqrt(0.01968133 + 0.06826667)
= sqrt(0.087948)
= 0.2962 (rounded to 4 decimal places)
The Z-score for a 95% confidence level is approximately 1.96.
Finally, we can substitute the values into the formula to calculate the upper limit:
Upper Limit = 0.18 + 1.96 * 0.2962
= 0.18 + 0.5791
= 0.7591 (rounded to 4 decimal places)
Therefore, the upper limit of a 95% confidence interval for the mean difference of the two populations is 0.7591.
To find the upper limit of a 95% confidence interval for the mean difference of the two populations, we can use the formula:
Upper limit = (Mean difference) + (Z-value) * (Standard deviation of the difference)
First, let's calculate the mean difference of the two populations:
Mean difference = Mean of specimens above the town - Mean of specimens below the town
= 4.92 - 4.74
= 0.18
Next, let's calculate the standard deviation of the difference:
Standard deviation of the difference = sqrt((Standard deviation above the town)^2 / (Sample size above the town) + (Standard deviation below the town)^2 / (Sample size below the town))
= sqrt((0.157^2 / 15) + (0.32^2 / 15))
≈ 0.133
Now, we need to find the Z-value for a 95% confidence interval. The Z-value represents the number of standard deviations away from the mean for a specific confidence level. For a 95% confidence interval, the Z-value is 1.96.
Finally, we can calculate the upper limit:
Upper limit = (0.18) + (1.96) * (0.133)
≈ 0.18 + 0.26168
≈ 0.44168
Therefore, the upper limit of a 95% confidence interval for the mean difference of the two populations is approximately 0.44168.
To calculate the upper limit of a 95% confidence interval for the mean difference of the two populations (above the town and below the town), you can follow these steps:
1. Determine the sample size of both groups:
- above the town: n₁ = 15
- below the town: n₂ = 15
2. Calculate the mean difference between the two groups:
- mean difference (d̄) = mean above the town - mean below the town
- d̄ = 4.92 - 4.74 = 0.18
3. Calculate the standard error of the mean difference:
- standard error (SE) = √((σ₁²/n₁) + (σ₂²/n₂))
- σ₁ = standard deviation above the town = 0.157
- σ₂ = standard deviation below the town = 0.32
- SE = √((0.157²/15) + (0.32²/15))
4. Determine the t-value for a 95% confidence interval with (n₁ + n₂ - 2) degrees of freedom:
- degrees of freedom (df) = n₁ + n₂ - 2 = 15 + 15 - 2 = 28
- t-value = t(0.025, df)
5. Calculate the margin of error (MOE):
- MOE = t-value * SE
6. Calculate the upper limit of the confidence interval for the mean difference:
- upper limit = d̄ + MOE
Following these steps, you can now calculate the answer.
Note: The "t(0.025, df)" represents the t-value at a significance level of 0.025 (2.5% in each tail) and with the given degrees of freedom. You can reference a t-distribution table or use statistical software to find the appropriate t-value.
Let's proceed with calculating the confidence interval.
Step 1:
n₁ = 15 (sample size above the town)
n₂ = 15 (sample size below the town)
Step 2:
d̄ = 4.92 - 4.74 = 0.18
Step 3:
SE = √((0.157²/15) + (0.32²/15))
SE ≈ 0.109
Step 4:
Find the t-value for a 95% confidence interval with 28 degrees of freedom:
t-value ≈ 2.048 (assuming from a t-distribution table or using software)
Step 5:
MOE = t-value * SE
MOE ≈ 2.048 * 0.109
MOE ≈ 0.223
Step 6:
Upper limit = d̄ + MOE
Upper limit = 0.18 + 0.223
Upper limit ≈ 0.403
Therefore, the upper limit of the 95% confidence interval for the mean difference of the two populations is approximately 0.403.