To answer this question, we need to use the central limit theorem, which states that the sampling distribution of the sample mean is approximately normal, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the population mean is 5 inches, the population standard deviation is 0.5 inches, and the sample size is 32. Therefore, the sampling distribution of the sample mean has a mean of 5 inches and a standard deviation of 0.5 / √32 ≈ 0.0884 inches.
We want to find the probability that the sample mean is greater than 4.8 inches, which is equivalent to finding the area under the normal curve to the right of 4.8 inches. To do this, we need to convert 4.8 inches to a standard score (z-score), which is the number of standard deviations away from the mean. The formula for the z-score is:
z = (x - μ) / σ
where x is the raw score, μ is the mean, and σ is the standard deviation. Plugging in the values, we get:
z = (4.8 - 5) / 0.0884 ≈ -2.2627
This means that 4.8 inches is about 2.26 standard deviations below the mean. To find the area to the right of this z-score, we can use a normal distribution calculator or a normal distribution table. Using the calculator, we enter the mean as 0, the standard deviation as 1, and the lower bound as -2.2627. The calculator gives us the area as 0.9883. This means that the probability of the sample mean being greater than 4.8 inches is 0.9883. Rounded to four decimal places, the answer is 0.9883.