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 lower limit of a 95% confidence interval for the mean differences of the two populations is:
To calculate the lower limit of the 95% confidence interval for the mean difference of the two populations, we can use the formula:
Lower limit = (mean of sample 1 - mean of sample 2) - (critical value * standard error)
First, we need to calculate the critical value. Since the sample size is small and we do not know the population standard deviation, we can use the t-distribution.
The degrees of freedom (DF) is calculated as (n1 + n2 - 2), where n1 and n2 are the sample sizes. In this case, the sample size for each group is 15, so the DF = (15 + 15 - 2) = 28.
To find the critical value, we need to look up the t-score at the 95% confidence level and with 28 degrees of freedom. We can use a t-table or a calculator to find this value. Let's assume it is 2.048 (rounded for simplicity).
Next, we need to calculate the standard error. The formula for the standard error of the mean difference is:
Standard error = sqrt[(s1^2 / n1) + (s2^2 / n2)]
where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
Plugging in the values:
Standard error = sqrt[(0.157^2 / 15) + (0.32^2 / 15)]
Calculating this, we get:
Standard error ≈ 0.1009
Now, we can calculate the lower limit of the confidence interval:
Lower limit = (4.92 - 4.74) - (2.048 * 0.1009)
Calculating this, we get:
Lower limit ≈ 0.18
Therefore, the lower limit of the 95% confidence interval for the mean difference of the two populations is approximately 0.18.
To find the lower limit of a 95% confidence interval for the mean difference of the two populations, we can use the formula:
Lower limit = (mean1 - mean2) - (t critical value * standard error)
First, let's calculate the standard error:
standard error = square root[(standard deviation1^2 / sample size1) + (standard deviation2^2 / sample size2)]
For the specimens above the town:
mean1 = 4.92
standard deviation1 = 0.157
sample size1 = 15
For the specimens below the town:
mean2 = 4.74
standard deviation2 = 0.32
sample size2 = 15
Calculating the standard error:
standard error = square root[(0.157^2 / 15) + (0.32^2 / 15)]
standard error ≈ 0.115
Next, we need the t critical value for a 95% confidence interval. Since the sample size is small (15), we need to use the t-distribution and degrees of freedom (df) = n1 + n2 - 2.
df = 15 + 15 - 2 = 28
Using a t-table or calculator, the t critical value for a 95% confidence level with 28 degrees of freedom is approximately 2.048.
Now, let's substitute the values into the formula:
Lower limit = (4.92 - 4.74) - (2.048 * 0.115)
Lower limit ≈ 0.18 - 0.235
Lower limit ≈ -0.055
Therefore, the lower limit of the 95% confidence interval for the mean difference of the two populations is approximately -0.055.
To find the lower limit of a 95% confidence interval for the mean differences of the two populations, you can follow these steps:
Step 1:
Calculate the standard error of the mean differences. This can be done using the formula:
Standard Error (SE) = √[(Standard Deviation1^2 / n1) + (Standard Deviation2^2 / n2)]
Where:
- Standard Deviation1 is the standard deviation of the first population (above the town)
- n1 is the sample size of the first population (above the town)
- Standard Deviation2 is the standard deviation of the second population (below the town)
- n2 is the sample size of the second population (below the town)
In this case:
Standard Deviation1 = 0.157
n1 = 15
Standard Deviation2 = 0.32
n2 = 15
Step 2:
Compute the margin of error (ME). This is done by multiplying the standard error (SE) by the critical value from the t-distribution at a 95% confidence level.
The critical value for a 95% confidence level with 14 degrees of freedom (15 samples - 1) can be found using a t-distribution table or calculator. For this example, the critical value is approximately 2.145.
Margin of Error (ME) = critical value * standard error (SE)
Step 3:
Calculate the lower limit of the confidence interval by subtracting the margin of error from the mean difference.
Lower Limit = Mean Difference - Margin of Error
In this case:
Mean Difference = Mean1 - Mean2
Mean1 = 4.92
Mean2 = 4.74
Using the values above, we can now calculate the lower limit.
Let's put it all together:
SE = √[(0.157^2 / 15) + (0.32^2 / 15)] ≈ 0.1471
ME = 2.145 * 0.1471 ≈ 0.3156
Lower Limit = 4.92 - 4.74 - 0.3156 ≈ -0.1356
Therefore, the lower limit of the 95% confidence interval for the mean differences of the two populations is approximately -0.1356.