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 be greater than the observed sample? (The sample means are female 167.2 with a sample size of 40 and male 179.41 with a sample sixe of 41)
To determine the probability of the difference between the mean male and female weights being greater than the observed sample, we can use the concept of the sampling distribution of the difference between means. Here's how you can calculate it:
Step 1: Calculate the standard deviation of the sampling distribution.
The standard deviation of the sampling distribution of the difference between means (also known as the standard error) can be calculated using the formula:
σd = √[(σ1^2 / n1) + (σ2^2 / n2)]
where σd is the standard deviation of the sampling distribution, σ1 and σ2 are the population standard deviations, and n1 and n2 are the sample sizes of each group.
Since the population standard deviations (σ1 and σ2) are not provided, we will assume that the sample standard deviations (s1 and s2) are good estimators of the population standard deviations. Hence, we will use the sample standard deviations to calculate the standard error:
σd = √[(s1^2 / n1) + (s2^2 / n2)]
In this case, the sample means and sample sizes are as follows:
Female: Sample mean = 167.2, Sample size = 40
Male: Sample mean = 179.41, Sample size = 41
Step 2: Calculate the z-score.
The z-score represents how many standard deviations the observed sample mean difference is away from the hypothesized population mean difference. It can be calculated using the formula:
z = (x̄1 - x̄2) / σd
where x̄1 and x̄2 are the sample means, and σd is the standard deviation of the sampling distribution.
Substituting the values into the formula:
z = (167.2 - 179.41) / σd
Step 3: Calculate the probability.
Finally, we can use the z-score to determine the probability associated with the observed sample mean difference being greater than the hypothesized population mean difference. We can consult a standard normal distribution (z-distribution) table or use statistical software to find this probability.
Note: Since you haven't provided the sample standard deviations, you will need to fill in those values to calculate the standard error and subsequently the z-score. Once you have both values, you can find the probability using the z-distribution table or software.
Remember, the probability will represent how likely it is to observe a sample mean difference as extreme as the one you observed in the given samples, assuming the populations from which the samples were drawn have the specific means provided.