1. The gain in weight of two random samples of patients fed on two different Diets: A and B are given below. Examine whether the difference in mean increase in weight is significant?Diet-A 13 14 10 11 2 16 10 8Diet-B 7 10 12 8 10 11 9 10 11
Question
- The gain in weight of two random samples of patients fed on two different Diets: A and B are given below. Examine whether the difference in mean increase in weight is significant?Diet-A 13 14 10 11 2 16 10 8Diet-B 7 10 12 8 10 11 9 10 11
Solution
To examine whether the difference in mean increase in weight between Diet A and Diet B is significant, we can perform a statistical test.
Step 1: Calculate the mean increase in weight for each diet. For Diet A, the mean increase in weight can be calculated by summing up all the weights and dividing by the number of samples: Mean_A = (13 + 14 + 10 + 11 + 2 + 16 + 10 + 8) / 8 = 11.5
For Diet B, the mean increase in weight can be calculated in the same way: Mean_B = (7 + 10 + 12 + 8 + 10 + 11 + 9 + 10 + 11) / 9 = 9.89
Step 2: Calculate the standard deviation for each diet. The standard deviation measures the variability of the data points around the mean. It can be calculated using the following formula:
Standard Deviation_A = sqrt(((13 - 11.5)^2 + (14 - 11.5)^2 + (10 - 11.5)^2 + (11 - 11.5)^2 + (2 - 11.5)^2 + (16 - 11.5)^2 + (10 - 11.5)^2 + (8 - 11.5)^2) / 8) = 4.06
Standard Deviation_B = sqrt(((7 - 9.89)^2 + (10 - 9.89)^2 + (12 - 9.89)^2 + (8 - 9.89)^2 + (10 - 9.89)^2 + (11 - 9.89)^2 + (9 - 9.89)^2 + (10 - 9.89)^2 + (11 - 9.89)^2) / 9) = 1.51
Step 3: Perform a t-test to determine if the difference in means is statistically significant. The t-test compares the means of two groups and determines if the difference between them is statistically significant. The formula for the t-test is:
t = (Mean_A - Mean_B) / sqrt((Standard Deviation_A^2 / n_A) + (Standard Deviation_B^2 / n_B))
Where n_A and n_B are the sample sizes for Diet A and Diet B, respectively.
In this case, n_A = 8 and n_B = 9.
t = (11.5 - 9.89) / sqrt((4.06^2 / 8) + (1.51^2 / 9)) = 1.14
Step 4: Determine the critical value and compare it with the calculated t-value. The critical value is determined based on the desired level of significance (e.g., 0.05). It represents the threshold beyond which we consider the difference in means to be statistically significant.
Using a t-table or statistical software, we can find the critical value for a two-tailed t-test with the appropriate degrees of freedom (df = n_A + n_B - 2). Let's assume the critical value is 2.04.
Step 5: Make a decision. If the calculated t-value is greater than the critical value, we reject the null hypothesis and conclude that the difference in means is statistically significant. If the calculated t-value is less than the critical value, we fail to reject the null hypothesis and conclude that the difference in means is not statistically significant.
In this case, the calculated t-value (1.14) is less than the critical value (2.04). Therefore, we fail to reject the null hypothesis and conclude that the difference in mean increase in weight between Diet A and Diet B is not statistically significant.
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