(b) The following set of lead determinations on separate aliquots of a pooled blood serum were reported: 103, 106, 107, and 114 meq/L. One value appears suspect. Should the suspected be retained or discarded at 95% confidence level.
It should be retained since there is no significance in the difference of the determinations.
Help me
To determine whether the suspected value should be retained or discarded at a 95% confidence level, we can use a t-test to assess its statistical significance. Here are the step-by-step instructions:
Step 1: Calculate the sample mean and sample standard deviation of the given determinations.
- The sample mean (x̄) is the sum of the individual values divided by the number of values.
- The sample standard deviation (s) measures the variability of the data points around the sample mean.
Given values: 103, 106, 107, 114
Sum of values: 103 + 106 + 107 + 114 = 430
Number of values: 4
Sample mean (x̄) = 430 / 4 = 107.5
To calculate the sample standard deviation, we need to calculate the sum of the squared differences between each value and the sample mean.
Step 2: Calculate the squared differences and sum them up.
- Subtract the sample mean from each individual value and square the result.
- Sum up all the squared differences.
Squared differences:
(103 - 107.5)^2 = 23.06
(106 - 107.5)^2 = 2.25
(107 - 107.5)^2 = 0.25
(114 - 107.5)^2 = 42.25
Sum of squared differences = 23.06 + 2.25 + 0.25 + 42.25 = 67.81
Step 3: Calculate the sample variance (s^2) by dividing the sum of squared differences by (n-1), where n is the number of values.
- Sample variance (s^2) = Sum of squared differences / (n-1)
Sample variance (s^2) = 67.81 / (4-1) = 67.81 / 3 ≈ 22.603
Step 4: Calculate the standard error of the sample distribution.
- Standard error = square root of (s^2 / n)
Standard error = √(22.603 / 4) ≈ √5.65075 ≈ 2.378
Step 5: Calculate the t-test statistic for the suspected value using the formula:
- t = (suspected value - x̄) / (standard error)
Given suspected value: 114
t = (114 - 107.5) / 2.378 ≈ 2.738
Step 6: Determine the critical t-value for a 95% confidence level.
- The critical t-value corresponds to the degrees of freedom (n-1) and the desired confidence level. In this case, the degrees of freedom is 3 (4-1).
To find the critical t-value, you can refer to a t-distribution table or use statistical software. For a 95% confidence level with 3 degrees of freedom, the critical t-value is approximately 3.182.
Step 7: Compare the t-test statistic to the critical t-value.
- If the absolute value of the t-test statistic is greater than the critical t-value, the suspected value is significantly different from the sample mean.
- If the absolute value of the t-test statistic is less than or equal to the critical t-value, the suspected value is not significantly different from the sample mean.
|2.738| ≤ 3.182
Since the absolute value of the t-test statistic (2.738) is less than or equal to the critical t-value (3.182), the suspected value of 114 should be retained at the 95% confidence level.
To determine whether the suspected value should be retained or discarded at a 95% confidence level, we can perform a statistical analysis called outlier detection. This analysis helps us identify values in a dataset that are significantly different from the others.
In this case, we have a set of lead determinations on separate aliquots of a pooled blood serum: 103, 106, 107, and 114 meq/L. We suspect that one value might be an outlier. To determine if it is indeed an outlier, we can calculate the mean and standard deviation of the dataset.
Step 1: Calculate the mean (average) of the dataset:
Mean = (103 + 106 + 107 + 114) / 4 = 430 / 4 = 107.5 meq/L
Step 2: Calculate the standard deviation of the dataset:
Standard deviation = √((∑(x - Mean)²) / (N - 1))
Where x represents each value in the dataset, and N is the number of values.
Substituting the values into the formula:
Standard deviation = √(((103 - 107.5)² + (106 - 107.5)² + (107 - 107.5)² + (114 - 107.5)²) / (4 - 1))
= √((17.5² + (-1.5)² + (-0.5)² + (6.5)²) / 3)
= √((306.25 + 2.25 + 0.25 + 42.25) / 3)
= √(351 / 3)
≈ √117
≈ 10.82 meq/L
Step 3: Calculate the z-score for the suspected value:
Z-score = (Suspected value - Mean) / Standard deviation
Z-score = (Suspected value - 107.5) / 10.82
In order to determine if the suspected value should be retained or discarded, we need to compare its z-score with the critical z-value at a 95% confidence level. The critical z-value for a 95% confidence level is approximately 1.96 (obtained from z-tables or statistical software).
If the calculated z-score is greater than 1.96 or smaller than -1.96, then the suspected value should be considered an outlier and discarded. Otherwise, it should be retained.
For example, let's say the suspected value is 120 meq/L. We can calculate the z-score as follows:
Z-score = (120 - 107.5) / 10.82 ≈ 1.16
Since the calculated z-score of 1.16 is smaller than 1.96, the suspected value should be retained at a 95% confidence level.
Note: It's important to mention that statistical analysis should be interpreted with caution, and in some cases, subject-matter expertise may be required to make a final decision based on the context and other factors.