To find the probability concerning the average weight of a sample of sixth graders, we can use the Central Limit Theorem. According to the theorem, the distribution of sample means will be normally distributed, regardless of the original distribution, provided the sample size is sufficiently large (which 50 is in this case).
Given that the average weight of a sixth grader is 80 lbs with a standard deviation of 20 lbs, we first need to determine the standard error (SE) of the sample mean. The standard error can be calculated using the formula:
SE = σ / √n
Where σ is the standard deviation of the population and n is the sample size. Plugging in our values:
SE = 20 / √50 ≈ 2.83
This means that the average weight of a sample of 50 sixth graders will have a standard deviation of approximately 2.83 lbs.
To find the probability of a sample mean being significantly different from 80 lbs, we would typically look for values that lie outside the range defined by a chosen significance level (e.g., 0.05 for a 95% confidence interval). For a two-tailed test, this range can be found using the means:
Lower Limit = μ - Z * SE
Upper Limit = μ + Z * SE
Where μ is the population mean (80 lbs), and Z is the Z-value corresponding to the desired level of confidence. For a 95% confidence level, the Z-value is approximately 1.96. Thus:
Lower Limit = 80 - 1.96 * 2.83 ≈ 74.44
Upper Limit = 80 + 1.96 * 2.83 ≈ 85.56
This gives us a range of approximately 74.44 lbs to 85.56 lbs. The probability of the sample mean being significantly different from 80 lbs would involve calculating the area under the normal curve that lies outside this range.
In conclusion, if we want to express the probability of drawing a sample mean that falls outside this range, we can look up the respective Z-scores for these limits and calculate the corresponding tail probabilities. Depending on the details of the specific question, such as how ‘significantly different’ is defined, you may end up with a specific probability value that reflects this deviation from the mean.