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.)
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.
To calculate the quartiles of the box weight, we need to use the information provided on the average weight and standard deviation.
First, let's find the z-scores for each quartile. The formula for the z-score is:
z = (x - μ) / σ
Where:
x = the value we want to find (the quartile)
μ = mean (average) weight
σ = standard deviation
We need to find the z-scores for the 25th, 50th, and 75th percentiles, which correspond to the first, second, and third quartiles.
For the 1st quartile:
z₁ = (x₁ - μ) / σ
For the 2nd quartile (median):
z₂ = (x₂ - μ) / σ
For the 3rd quartile:
z₃ = (x₃ - μ) / σ
Now let's plug in the values:
μ = 19 ounces (mean weight)
σ = 0.6 ounce (standard deviation)
For the 1st quartile (25th percentile), the z-score is -0.67449 (found using a z-table or any statistical software).
z₁ = (x₁ - 19) / 0.6
Now solve for x₁:
-0.67449 = (x₁ - 19) / 0.6
Multiply both sides by 0.6:
-0.67449 * 0.6 = x₁ - 19
-0.404694 = x₁ - 19
x₁ = -0.404694 + 19
x₁ = 18.595306
So, the 1st quartile of the box weight is approximately 18.60 ounces (rounded to 2 decimal places).
For the 2nd quartile (50th percentile), the z-score is 0 (since it's the median of a normal distribution).
z₂ = (x₂ - 19) / 0.6
0 = (x₂ - 19) / 0.6
Multiply both sides by 0.6:
0 * 0.6 = x₂ - 19
0 = x₂ - 19
x₂ = 19
So, the 2nd quartile (median) of the box weight is exactly 19 ounces.
For the 3rd quartile (75th percentile), the z-score is 0.67449.
z₃ = (x₃ - 19) / 0.6
0.67449 = (x₃ - 19) / 0.6
Multiply both sides by 0.6:
0.67449 * 0.6 = x₃ - 19
0.404694 = x₃ - 19
x₃ = 0.404694 + 19
x₃ = 19.404694
So, the 3rd quartile of the box weight is approximately 19.40 ounces (rounded to 2 decimal places).
In summary:
1st quartile: 18.60 ounces
2nd quartile (median): 19 ounces
3rd quartile: 19.40 ounces
To calculate the quartiles of the box weight, we need to convert the weights to standard normal distribution using z-scores. The formula for calculating the z-score is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the value we want to convert
- μ is the mean of the distribution (average weight)
- σ is the standard deviation
In this case, the average weight (mean) is 19 ounces and the standard deviation is 0.6 ounce.
Let's find the z-scores for the quartiles:
1st Quartile:
The 1st quartile represents the 25th percentile, which means 25% of the weights will be below this value. We can calculate the z-score for the 25th percentile using the z-table or a statistical calculator.
Using the z-table, we look for the value closest to 0.25 in the body of the table. We find that z = -0.674.
Now we can use the z-score formula to calculate the corresponding weight:
z = (x - μ) / σ
-0.674 = (x - 19) / 0.6
Solving for x:
x = -0.674 * 0.6 + 19
x ≈ 18.6
Therefore, the 1st quartile of the box weight is approximately 18.6 ounces.
2nd Quartile (Median):
The 2nd quartile represents the 50th percentile, which is the median. We do not need to calculate the z-score for the median since it is 0 in the standard normal distribution.
So, the 2nd quartile (median) of the box weight is the same as the average weight, which is 19 ounces.
3rd Quartile:
The 3rd quartile represents the 75th percentile, which means 75% of the weights will be below this value. Again, we can use the z-table to find the z-score closest to 0.75, which is 0.675.
Using the z-score formula, we can calculate the corresponding weight:
z = (x - μ) / σ
0.675 = (x - 19) / 0.6
Solving for x:
x = 0.675 * 0.6 + 19
x ≈ 19.4
Therefore, the 3rd quartile of the box weight is approximately 19.4 ounces.
In summary, the 1st quartile is approximately 18.6 ounces, the 2nd quartile (median) is 19 ounces, and the 3rd quartile is approximately 19.4 ounces.