In a harvest of pumpkins, the mean diameter of the pumpkins is 21 inches with a standard deviation of 5 inches. The diameters are normally distributed, so 95% of the pumpkins fall within a certain range of values around the mean. Select the pumpkins that fall into this range.
To find the range of values in which 95% of the pumpkins fall, we need to determine the z-score corresponding to the 95th percentile.
First, we find the z-score using the z-table or calculator. The area to the left of this z-score represents the percentage of pumpkins falling within the range.
Using the z-table, we find that the z-score for a cumulative probability of 0.95 is approximately 1.96 (rounded to two decimal places).
Next, we calculate the range of values around the mean by adding and subtracting the product of the standard deviation and the z-score from the mean:
Lower limit = Mean - (Standard Deviation * z-score)
Lower limit = 21 - (5 * 1.96)
Lower limit = 21 - 9.8
Lower limit ≈ 11.2
Upper limit = Mean + (Standard Deviation * z-score)
Upper limit = 21 + (5 * 1.96)
Upper limit = 21 + 9.8
Upper limit ≈ 30.8
Therefore, 95% of the pumpkins fall within the range of approximately 11.2 to 30.8 inches in diameter.