位置15的问题 A population has a mean of 84 and a standard deviation of 12. A sample of 36 observations will be taken. The probability that the sample mean will be between 80.54 and 88.9 is 题库 1/1[input]. (Note: your answer should retain THREE decimal places.)

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BlackTom AI · Accepted Answer

The problem asks for the probability that the sample mean falls between 80.54 and 88.9, given a population mean of 84, population standard deviation of 12, and sample size n = 36.
First, identify the sampling distribution: since the population is normal or the sample size is large, the sample mean  X-bar  is approximately normal with mean μ = 84 and standard error σ¯ = σ/√(n) = 12/6 = 2.
Next, convert the interval endpoints to z-scores:
- z1 = (80.54 - 84) / 2 = -3.46 / 2 = -1.73
- z2 = (88.9 - 84) / 2 = 4.9 / 2 = 2.45
Then, use the standard normal distribution to find the corresponding probabilities:
- P(Z ≤ -1.73) = Phi(-1.73) = 1 - Phi(1.73) ≈ 1 - 0.9582 ≈ 0.0418
- P(Z ≤ 2.45) = Phi(2.45) ≈ 0.9929
Finally, the probability that the sample mean lies between 80.54 and 88.9 is:
P(80.54 ≤ X-bar ≤ 88.9) = Phi(2.45) − Phi(-1.73) ≈ 0.9929 − 0.0418 ≈ 0.9511.
Rounding to three decimal places yields 0.951.
If you prefer, you can think of this as the area between -1.73 and 2.45 under the standard normal curve, which aligns with the same calculation steps and results.

位置15的问题 A population has a mean of 84 and a standard deviation of 12. A sample of 36 observations will be taken. The probability that the sample mean will be between 80.54 and 88.9 is 题库 1/1[input]. (Note: your answer should retain THREE decimal places.)

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