Final answer:
To find the probability that the sample mean is within 0.5 of the population mean, calculate the standard error (standard deviation of the sampling distribution of the sample mean), convert the margin to z-scores, and consult a z-table or calculator for the probability.
Step-by-step explanation:
The question asks for the probability that the sample mean x will be within 0.5 of the population mean for a sample of size 36 from a population with mean 47 and standard deviation 8. To solve this, we use the formula for the standard deviation of the sampling distribution of the sample mean, which is σ/√n, where σ is the population standard deviation and n is the sample size. In this case, the standard deviation of the sampling distribution, also known as the standard error, is 8/√36 = 8/6 = 1.33.
We then convert the 0.5 margin to z-scores by dividing by the standard error, resulting in z-scores of ±0.5/1.33 ≈ ±0.38. Consulting a z-table or using a calculator for the standard normal distribution, we find the probability corresponding to these z-scores.
The exact probabilities will come from finding the area under the standard normal curve between the z-scores of -0.38 and +0.38. As the z-scores are close to 0, which is the mean of the standard normal distribution, the probability that the sample mean will be within 0.5 of the population mean is expected to be quite high, but to find the precise probability, we'd refer to the normal distribution tables or a calculator.