Calcium is essential to tree growth because it promotes the formation of wood and maintains cell walls. In 1990, the concentration of calcium in precipitation in a certain area was 0.15 milligrams per liter (mg/L). A random sample of 10 precipitation dates in 2007 results in the following data table.
Construct a 95% confidence interval about the sample mean concentration of calcium precipitation.
Sample Mean = 0.19 mg/L
95% Confidence Interval = (0.14, 0.24) mg/L
To construct a 95% confidence interval about the sample mean concentration of calcium precipitation, we can follow these steps:
1. Determine the sample mean (x̄) of the calcium concentration. This is calculated by summing up all the values in the sample and dividing it by the sample size (n). In this case, we would sum up the 10 values from 2007 and divide it by 10.
2. Calculate the sample standard deviation (s) of the calcium concentration. This measures the variability of the data points in the sample. To find the sample standard deviation, subtract the sample mean from each data point, square the differences, sum up these squared differences, divide by (n-1), and finally take the square root.
3. Determine the critical value for a 95% confidence level. The critical value is obtained from a t-distribution table based on the sample size (n-1) and desired level of confidence. In this case, with a sample size of 10, the degrees of freedom is 9. From the t-distribution table, the critical value for a 95% confidence interval with 9 degrees of freedom is approximately 2.262.
4. Calculate the margin of error. The margin of error represents the maximum amount of error that can be tolerated in our point estimate. It is calculated by multiplying the critical value from step 3 with the sample standard deviation divided by the square root of the sample size (s/√n).
5. Construct the confidence interval. The confidence interval is calculated by subtracting the margin of error from the sample mean to obtain the lower bound, and adding the margin of error to the sample mean to obtain the upper bound.
With these steps, we can construct the 95% confidence interval about the sample mean concentration of calcium precipitation using the provided data.
To construct a 95% confidence interval about the sample mean concentration of calcium precipitation, you need to calculate the mean and standard deviation of the sample and use these values to determine the interval.
Here are the precipitation data values for the random sample of 10 dates in 2007:
0.12 mg/L
0.18 mg/L
0.21 mg/L
0.14 mg/L
0.16 mg/L
0.13 mg/L
0.15 mg/L
0.19 mg/L
0.17 mg/L
0.11 mg/L
Step 1: Calculate the sample mean (x̄):
x̄ = (0.12 + 0.18 + 0.21 + 0.14 + 0.16 + 0.13 + 0.15 + 0.19 + 0.17 + 0.11) / 10 = 0.155 mg/L
Step 2: Calculate the sample standard deviation (s):
s = sqrt((∑(x - x̄)^2) / (n - 1))
= sqrt(((0.12 - 0.155)^2 + (0.18 - 0.155)^2 + (0.21 - 0.155)^2 + (0.14 - 0.155)^2 + (0.16 - 0.155)^2 + (0.13 - 0.155)^2 + (0.15 - 0.155)^2 + (0.19 - 0.155)^2 + (0.17 - 0.155)^2 + (0.11 - 0.155)^2) / (10 - 1))
= sqrt((0.000025 + 0.000025 + 0.000256 + 0.000036 + 0.000009 + 0.000025 + 0.000000 + 0.000064 + 0.000064 + 0.000469) / 9)
= sqrt(0.000494 / 9)
= sqrt(0.000055)
= 0.0074 mg/L
Step 3: Determine the critical value for a 95% confidence interval.
Since the sample size (n) is less than 30, you can use the t-distribution. For a 95% confidence level with 9 degrees of freedom (n-1), the critical value is 2.262 from the t-distribution table.
Step 4: Calculate the margin of error (E):
E = Critical value * (s / sqrt(n))
= 2.262 * (0.0074 / sqrt(10))
= 0.0052 mg/L
Step 5: Calculate the lower and upper bound of the confidence interval:
Lower bound = x̄ - E
= 0.155 - 0.0052
= 0.1498 mg/L
Upper bound = x̄ + E
= 0.155 + 0.0052
= 0.1602 mg/L
Therefore, the 95% confidence interval about the sample mean concentration of calcium precipitation is (0.1498 mg/L, 0.1602 mg/L).