Samples of both surface soil and subsoil were taken from eight randomly selected farms in a county. The soil samples were analyzed to determine both surface pH and subsoil pH, which yielded the following results:
Farm
1 2 3 4
Surface pH: 6.55 5.98 5.59 6.17
Subsoil pH: 6.78 6.14 5.80 5.91
5 6 7 8
Surface pH: 5.92 6.18 6.43 5.68
Subsoil pH: 6.10 6.01 6.18 5.88
a. Compute a 90% confidence interval for the true average difference between surface and subsoil pH for farmland in this county.
b. What assumptions are necessary to validate the interval in Part a?
Please help with any info./formula/calculator notation... Anything!
To compute a confidence interval for the true average difference between surface and subsoil pH, we can use the following formula:
CI = (x̄d - tα/2 * s / √n, x̄d + tα/2 * s / √n)
Where:
- CI represents the confidence interval
- x̄d represents the average difference between surface and subsoil pH
- tα/2 represents the critical value from the t-distribution for the desired confidence level, α/2
- s represents the standard deviation of the differences between surface and subsoil pH
- n represents the number of samples
Now, let's calculate the confidence interval:
Step 1: Calculate the average difference (x̄d)
First, calculate the difference between surface and subsoil pH for each farm, and then find the average difference.
Farm 1: 6.55 - 6.78 = -0.23
Farm 2: 5.98 - 6.14 = -0.16
Farm 3: 5.59 - 5.80 = -0.21
Farm 4: 6.17 - 5.91 = 0.26
Farm 5: 5.92 - 6.10 = -0.18
Farm 6: 6.18 - 6.01 = 0.17
Farm 7: 6.43 - 6.18 = 0.25
Farm 8: 5.68 - 5.88 = -0.20
x̄d = (-0.23 - 0.16 - 0.21 + 0.26 - 0.18 + 0.17 + 0.25 - 0.20) / 8 = -0.0125
Step 2: Calculate the standard deviation (s)
Next, calculate the standard deviation of the differences between surface and subsoil pH.
s = √[(Σ(xd - x̄d)²) / (n - 1)]
xd represents the difference between surface and subsoil pH for each farm.
Σ(xd - x̄d)² = (-0.23 - (-0.0125))² + (-0.16 - (-0.0125))² + (-0.21 - (-0.0125))² + (0.26 - (-0.0125))² + (-0.18 - (-0.0125))² + (0.17 - (-0.0125))² + (0.25 - (-0.0125))² + (-0.20 - (-0.0125))² = 0.3383
s = √(0.3383 / (8 - 1)) = 0.206
Step 3: Determine the critical value (tα/2)
The critical value depends on the desired confidence level and degrees of freedom. Since we are calculating a 90% confidence interval, the confidence level is 0.9, and the degrees of freedom are (n - 1) = 8 - 1 = 7.
Using a t-distribution table or a calculator, the critical value for a 90% confidence level and 7 degrees of freedom is approximately 1.894.
Step 4: Calculate the confidence interval (CI)
Substituting the values into the confidence interval formula:
CI = (-0.0125 - 1.894 * 0.206 / √8, -0.0125 + 1.894 * 0.206 / √8)
CI = (-0.221, 0.196)
Therefore, the 90% confidence interval for the true average difference between surface and subsoil pH for farmland in this county is (-0.221, 0.196).
Now, let's move on to part b to discuss the assumptions necessary to validate this interval.
To compute a confidence interval for the true average difference between surface and subsoil pH for farmland in this county, we can use the formula for the confidence interval of the difference between two means.
a. Formula for the confidence interval for the difference between two means:
CI = (x̄₁ - x̄₂) ± (t * SE)
where:
- CI is the confidence interval
- x̄₁ is the sample mean of the first group (surface pH)
- x̄₂ is the sample mean of the second group (subsoil pH)
- t is the critical value from the t-distribution based on the desired confidence level and degrees of freedom
- SE is the standard error of the difference
First, calculate the sample means for the surface and subsoil pH:
- Surface pH sample mean (x̄₁) = (6.55 + 5.98 + 5.59 + 6.17 + 5.92 + 6.18 + 6.43 + 5.68) / 8
- Subsoil pH sample mean (x̄₂) = (6.78 + 6.14 + 5.80 + 5.91 + 6.10 + 6.01 + 6.18 + 5.88) / 8
Next, calculate the standard deviation (s) for the differences between the surface and subsoil pH within each farm:
- Calculate the difference between surface pH and subsoil pH for each farm: (6.55 - 6.78), (5.98 - 6.14), (5.59 - 5.80), ...
- Compute the sample standard deviation of these differences
Then, calculate the standard error of the difference (SE):
- SE = s / √n
- s is the standard deviation of the differences
- n is the number of observations (equal in both groups)
Next, determine the critical value (t) based on the desired confidence level and degrees of freedom:
a. For a 90% confidence level, a two-tailed t-distribution with seven degrees of freedom (n₁ + n₂ - 2 = 8 + 8 - 2 = 14) has a critical value of approximately 1.761.
Finally, plug in the values into the formula for the confidence interval (CI):
CI = (x̄₁ - x̄₂) ± (t * SE)
b. Assumptions necessary to validate the interval:
To validate the confidence interval, the following assumptions should be met:
- The soil samples were taken randomly from the farms in the county.
- The differences between surface and subsoil pH are normally distributed.
- The surface and subsoil measurements are independent of each other.
- The variances of the surface and subsoil pH measurements are equal.
Once you have the values from the calculations, you can use a calculator or software (such as Microsoft Excel or statistical packages like R) to perform the calculations and get the final confidence interval for the true average difference between surface and subsoil pH for farmland in this county.