To determine the sampling distribution of the sample mean for a random sample of size \( n=2 \) years from the given population distribution of shipwrecks, we need to follow these steps:
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Understand the Population Distribution: We have the following probabilities for different ranges of shipwrecks:
- \( P(0 \leq x < 0.5) = 0.2 \)
- \( P(0.5 \leq x < 1.5) = 0.5 \)
- \( P(1.5 \leq x < 3) = 0.2 \)
Note that the mean of this population has been given as part of the problem, though you didn't specify its value. For the sake of this analysis, let's denote the population mean as \( \mu \).
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Calculate Possible Sample Means: Since we are taking samples of size \( n=2 \), we can have different combinations of shipwrecks sampled over the two years. We can summarize this as follows:
- Possible mean for combinations:
- \( (x_1, x_2) \) where each \( x_i \) can take values within the ranges determined by the probability histogram.
- Possible mean for combinations:
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Calculate Sampling Distribution: We’ll need to consider the probabilities of the combinations of possible shipwreck counts. The sample mean is computed as \( \bar{x} = \frac{x_1 + x_2}{2} \).
Here are the combinations we'll consider:
- From the interval \( [0, 0.5) \):
- \( (0, 0) \), \( (0, 0.5) \), \( (0.5, 0) \), \( (0.5, 0.5) \)
- From the interval \( [0.5, 1.5) \):
- \( (0.5, 0) \), \( (0.5, 0.5) \), etc.
As you calculate sample means for all the possible pairs from the ranges, you will end up with the following values for means (noting that you will ultimately weight these based on their probabilities):
- From the interval \( [0, 0.5) \):
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Assign Probabilities to Sample Means: To find the sampling distribution, we would calculate the probabilities associated with each unique sample mean value and then create a new distribution based on these probabilities.
For instance, if \( x_1 \) and \( x_2 \) both fall in the range \([0, 0.5)\), the sample mean will be in that range as hinted above. Similarly, a sample mean could be in the range \([0.5, 1.5)\) and so on.
- Final Distribution: Collect sample mean values with their corresponding probabilities and build the probability distribution of the sample mean.
To summarize, while I cannot give you the exact numbers without knowing the mean value or having clarity on integrating ranges, the procedure will involve listing out possible combinations of shipwreck counts, calculating their means, and determining the associated probabilities based on the original distribution.
A useful approximation, as you sample larger \( n \), is that by the Central Limit Theorem, as \( n \) increases, the distribution of the sample mean will approach a normal distribution with mean \( \mu \) and a variance of \( \sigma^2 / n\), where \( \sigma^2 \) is the variance of the population distribution. However, take caution that for small \( n \), such as 2, this normal approximation is less accurate unless the original population distribution is already approximately normal.