To calculate the quartiles, we need to find the z-scores corresponding to each quartile and then use the formula z = (x - μ) / σ to find the corresponding weights.
1. First Quartile (Q1):
The first quartile corresponds to the 25th percentile. We can find the z-score for the 25th percentile using the z-table.
From the z-table, the z-score for the 25th percentile is approximately -0.6745.
Using the formula z = (x - μ) / σ, we can rearrange it to find x:
x = z * σ + μ
x = -0.6745 * 0.6 + 19
x ≈ 18.5953
Round to 2 decimal places:
Q1 ≈ 18.60 ounces
2. Second Quartile (Q2) or Median:
The second quartile corresponds to the 50th percentile, which is also the median of a normal distribution.
Using the formula x = z * σ + μ, for the median, the z-score is 0.
x = 0 * 0.6 + 19
x = 19
Q2 = Median = 19 ounces
3. Third Quartile (Q3):
The third quartile corresponds to the 75th percentile. We can find the z-score for the 75th percentile using the z-table.
From the z-table, the z-score for the 75th percentile is approximately 0.6745.
Using the formula x = z * σ + μ, we can find x:
x = 0.6745 * 0.6 + 19
x ≈ 19.4047
Round to 2 decimal places:
Q3 ≈ 19.40 ounces
Therefore, the first quartile (Q1) is approximately 18.60 ounces, the second quartile (Q2) or median is 19 ounces, and the third quartile (Q3) is approximately 19.40 ounces.
A packaging system fills boxes to an average weight of 19 ounces with a standard deviation of 0.6 ounce. It is reasonable to assume that the weights are normally distributed. Calculate the 1st, 2nd, and 3rd quartiles of the box weight. (You may find it useful to reference the z table. Round your final answers to 2 decimal places.)
1 answer
1 answer