Answer:
The value of μ is approximately 258.6
Step-by-step explanation:
To find the value of μ, we need to use the formula for the standard error of the mean (σxˉ):
σxˉ = σ / √n
Where σ is the population standard deviation and n is the sample size.
Given that σxˉ = 9.31 and n = 576, we can rearrange the formula to solve for σ:
σ = σxˉ * √n
σ = 9.31 * √576
σ = 9.31 * 24
σ = 223.44
Now, we can use the Z-score formula to find the Z-score corresponding to the given percentage:
Z = (x - μ) / (σ / √n)
In this case, x is 269.3996, and the percentage is 12.3%. We need to find the Z-score that corresponds to the upper tail of the distribution, so we subtract the percentage from 100%:
Z = (269.3996 - μ) / (223.44 / √576)
To find the Z-score corresponding to 100% - 12.3% = 87.7%, we can use a Z-table or a calculator. The Z-score that corresponds to 87.7% is approximately 1.16.
1.16 = (269.3996 - μ) / (223.44 / √576)
Now, we can solve for μ:
1.16 * (223.44 / √576) = 269.3996 - μ
(1.16 * 223.44) / √576 = 269.3996 - μ
(1.16 * 223.44) / √576 + μ = 269.3996
μ = 269.3996 - (1.16 * 223.44) / √576
μ ≈ 258.6
Therefore, the value of μ is approximately 258.6.