Consider a hypothetical population of dogs in which there are four possible weights, all of which are equally likely: 42, 48, 52, or 58 pounds. If a sample of size n=2 is drawn from this population, what is the sampling distribution of the total weight of the two dogs selected? That is, what the possible values for the total and what are the probabilities associated with each of those values?
noooooooooooooooooooooooooooooooooooo
i don't know please give me the answer
To determine the sampling distribution of the total weight of two dogs selected, we can consider all possible combinations of weights and calculate the probability for each combination.
Let's denote the weights as A, B, C, and D, where A = 42 pounds, B = 48 pounds, C = 52 pounds, and D = 58 pounds.
The possible combinations of weights for the two dogs are (in no particular order):
1. A + A = 42 + 42 = 84 pounds
2. A + B = 42 + 48 = 90 pounds
3. A + C = 42 + 52 = 94 pounds
4. A + D = 42 + 58 = 100 pounds
5. B + B = 48 + 48 = 96 pounds
6. B + C = 48 + 52 = 100 pounds
7. B + D = 48 + 58 = 106 pounds
8. C + C = 52 + 52 = 104 pounds
9. C + D = 52 + 58 = 110 pounds
10. D + D = 58 + 58 = 116 pounds
Since each weight is equally likely, we can calculate the probability for each combination by dividing the number of times each combination occurs by the total number of possible combinations.
There are a total of 4 possible weights, and we are drawing 2 dogs from the population (n=2), so the total number of combinations is given by the combination formula:
Total combinations = (4 choose 2) = 4!/(2!(4-2)!) = 6
Now, let's calculate the probabilities for each combination:
1. P(A + A) = 1/6
2. P(A + B) = 1/6
3. P(A + C) = 1/6
4. P(A + D) = 1/6
5. P(B + B) = 1/6
6. P(B + C) = 1/6
7. P(B + D) = 1/6
8. P(C + C) = 1/6
9. P(C + D) = 1/6
10. P(D + D) = 1/6
Therefore, the sampling distribution of the total weight of the two dogs selected is as follows:
Total weight = 84 pounds, Probability = 1/6
Total weight = 90 pounds, Probability = 1/6
Total weight = 94 pounds, Probability = 1/6
Total weight = 100 pounds, Probability = 1/6
Total weight = 96 pounds, Probability = 1/6
Total weight = 100 pounds, Probability = 1/6
Total weight = 106 pounds, Probability = 1/6
Total weight = 104 pounds, Probability = 1/6
Total weight = 110 pounds, Probability = 1/6
Total weight = 116 pounds, Probability = 1/6
To determine the sampling distribution of the total weight of the two dogs selected, we need to consider all possible combinations of weights that can be drawn from the population.
Given that there are four possible weights (42, 48, 52, and 58 pounds) and that the sample size is n=2, we can calculate all the possible values for the total weight by enumerating all the combinations.
Let's list all the possible combinations:
- Combination 1: 42 + 42 = 84
- Combination 2: 42 + 48 = 90
- Combination 3: 42 + 52 = 94
- Combination 4: 42 + 58 = 100
- Combination 5: 48 + 48 = 96
- Combination 6: 48 + 52 = 100
- Combination 7: 48 + 58 = 106
- Combination 8: 52 + 52 = 104
- Combination 9: 52 + 58 = 110
- Combination 10: 58 + 58 = 116
Now, let's calculate the probabilities associated with each of these combinations. Remember that all the weights are equally likely, so each combination has an equal probability.
There are a total of 10 combinations, and since each combination is equally likely, the probability associated with each combination is 1/10.
Therefore, we have the following sampling distribution of the total weight and their associated probabilities:
- Total weight: 84 pounds, Probability: 1/10
- Total weight: 90 pounds, Probability: 1/10
- Total weight: 94 pounds, Probability: 1/10
- Total weight: 100 pounds, Probability: 1/10 (Two combinations result in this total weight)
- Total weight: 96 pounds, Probability: 1/10
- Total weight: 106 pounds, Probability: 1/10
- Total weight: 104 pounds, Probability: 1/10
- Total weight: 110 pounds, Probability: 1/10
- Total weight: 116 pounds, Probability: 1/10
In summary, the sampling distribution of the total weight of the two dogs selected from the given population consists of the above possible values and their associated probabilities.