Final answer:
The 10th percentile of pit depths is approximately 780 μm. A pit depth of 780 μm is on the 9th percentile. The proportion of pits with depths between 800 and 830 μm is approximately 40.14%.
Step-by-step explanation:
To find the 10th percentile of pit depths, we can use the z-score formula. First, we calculate the z-score using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. In this case, x = 818, μ = 818, and σ = 29. Plugging these values in, we get z = (818 - 818) / 29 = 0. To find the corresponding percentile, we can look up the z-score in a z-table. The 10th percentile corresponds to a z-score of approximately -1.2816. To find the value, we can use the formula z = (x - μ) / σ and solve for x. Rearranging the formula, we have x = μ + zσ = 818 + (-1.2816) * 29 = 780.35 ≈ 780. Therefore, the 10th percentile of pit depths is approximately 780 μm.
To find the percentile that a certain pit depth of 780 μm is on, we can use the z-score formula and the same steps as before. Plugging in the values, we get z = (780 - 818) / 29 = -1.3103. Looking up this z-score in the z-table, we find that it corresponds to a percentile of approximately 9%. Therefore, the pit depth of 780 μm is on the 9th percentile.
To find the proportion of pits with depths between 800 and 830 μm, we can first find the z-scores for these values using the same formula. Plugging in the values, we get z1 = (800 - 818) / 29 = -0.6207 and z2 = (830 - 818) / 29 = 0.4138. To find the proportion between these z-scores, we can subtract the cumulative area corresponding to z1 from the cumulative area corresponding to z2. Looking up these z-scores in the z-table, we find that the cumulative areas are approximately 26.41% and 66.55%, respectively. Therefore, the proportion of pits with depths between 800 and 830 μm is approximately 66.55% - 26.41% = 40.14%.