Answer: To find the probability that the mean length of 39 items is greater than 5.8 inches, we can use the central limit theorem, which states that the distribution of sample means of a large enough sample will be approximately normally distributed, regardless of the shape of the original population.
The mean of the sample means will be equal to the mean of the original population, which is 6.4 inches.
The standard deviation of the sample means (standard error) can be calculated using the formula:
Standard error = standard deviation of the population / sqrt(sample size)
Standard error = 1.8 / sqrt(39) ≈ 0.2906
Now, we want to find the probability that the mean length of 39 items is greater than 5.8 inches. We can convert this to a z-score using the formula:
z = (x - μ) / SE
where x is the value we want to find the probability for (5.8 inches), μ is the population mean (6.4 inches), and SE is the standard error (0.2906).
z = (5.8 - 6.4) / 0.2906 ≈ -2.0617
Now, we need to find the probability that the z-score is greater than -2.0617. We can look up this probability in a standard normal distribution table or use a calculator.
The probability that the mean length of 39 items is greater than 5.8 inches is approximately 0.9796.
Therefore, the probability is 0.9796 (rounded to four decimal places).