If these samples were drawn from a population with mean male weight of 180 and mean female weight of 165, what is the probability that the difference between the mean male and female weights would lie between 4.5 and 27.8?

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To calculate the probability that the difference between the mean male and female weights lies between 4.5 and 27.8, we need additional information about the population standard deviation and the sample size for each group.

Assuming both groups have equal sample sizes (n) and equal standard deviations (σ) for simplicity, we can use the formula for the difference between two sample means:

Standard Error of the Difference (SE) = σ * √(2/n)

where σ is the population standard deviation and n is the sample size of each group.

Once we have the standard error, we can convert the difference between the mean male and female weights (in this case, 4.5 and 27.8) into a z-score using the formula:

z = (x - μ) / SE

where x represents the difference between the mean male and female weights, and μ is the hypothesized difference (in this case, 180 - 165).

By calculating the z-scores for both values (4.5 and 27.8), we can then look up the corresponding areas under the standard normal distribution curve (z-table) to find the probability.