Duplicate reading, using two separate instruments, were obtained on ten samples. What is the tabular value of the statistic used to determine the significance of the mean difference at the 0.05 level of probability?
The correct answer and explanation is:
To determine the significance of the mean difference between two separate instruments, we typically use a paired t-test when dealing with paired or duplicate readings.
Given that the samples are paired, we can calculate the mean difference between each pair and then assess the significance of this difference. The paired t-test statistic is calculated as: t=dˉsd/nt = \frac{\bar{d}}{s_d / \sqrt{n}}
Where:
- dˉ\bar{d} is the mean of the differences (i.e., the difference between the paired readings for each sample).
- sds_d is the standard deviation of the differences.
- nn is the number of pairs (in this case, 10 pairs).
For the hypothesis test, we will test:
- Null Hypothesis (H0H_0): The mean difference is zero (no significant difference between the instruments).
- Alternative Hypothesis (HAH_A): The mean difference is not zero (there is a significant difference between the instruments).
To find the tabular value of the statistic (also known as the critical value), we need to use the t-distribution table. The critical value depends on the degrees of freedom (df), which for a paired t-test is calculated as df=n−1df = n – 1.
In this case, n=10n = 10, so the degrees of freedom are: df=10−1=9df = 10 – 1 = 9
Now, we check the t-distribution table for df=9df = 9 at the 0.05 significance level (for a two-tailed test). For this setup, the critical value from the t-table is approximately 2.262.
This value represents the cut-off point. If the calculated t-statistic exceeds this value in absolute terms, we reject the null hypothesis and conclude that there is a significant difference between the two instruments. If the calculated t-statistic is smaller than this value, we fail to reject the null hypothesis, implying no significant difference.
Summary:
- The tabular value of the t-statistic for 9 degrees of freedom at the 0.05 level of significance is 2.262.
- To interpret this result, you would compare the calculated t-statistic to this critical value to determine if the mean difference is statistically significant.