The masses of boxes of apples are normally distributed such that 20% of the boxes are heavier than 5.08 kg and 15% of the boxes are heavier than 5.62kg. Estimate the mean and standard deviation of the masses.
let the mean be m, and the standard deviation be s
(5.08-m)/s < .8
using the equality: m + .8s = 5.08
(5.62-m) < .85s --> m + .85s = 5.62
subtract them: .05s = .54
s = 10.8
m + .8(10.8) = 5.08
m = negative, uh uh
rethinking this question, using ...
http://davidmlane.com/normal.html
Clicking on "value from an area", I entered .8 and got
a z-score of .841 , clicking "above"
entering .85 , my z-score was 1.036 clicking above
(5.08 - m)/s = .841 ---> m + .841s = 5.08
(5.62 - m)/s = 1.036 --- m + 1.036s = 5.62
subtract : .195s = .54
s = 2.77 , then m = 2.75
To estimate the mean and standard deviation of the masses of boxes of apples, we can use the standard normal distribution.
Let's denote the mean as μ and the standard deviation as σ.
Step 1: Calculate the Z-score for the first condition where 20% of the boxes are heavier than 5.08 kg.
Using the standard normal distribution table or calculator, we can find the Z-score corresponding to a cumulative probability of 0.20. The Z-score is approximately -0.84.
Step 2: Calculate the Z-score for the second condition where 15% of the boxes are heavier than 5.62 kg.
Using the standard normal distribution table or calculator, we can find the Z-score corresponding to a cumulative probability of 0.15. The Z-score is approximately -1.04.
Step 3: Use the Z-scores to set up two equations.
Since we know that the values corresponding to the Z-scores are the masses, we can set up two equations:
Z1 = (5.08 - μ) / σ = -0.84
Z2 = (5.62 - μ) / σ = -1.04
Step 4: Solve the equations for μ and σ.
From equation 1, we can rearrange and solve for μ:
μ = 5.08 - Z1 * σ
From equation 2, we can rearrange and solve for σ:
σ = (5.62 - μ) / Z2
Substituting the value of μ from equation 1 into equation 2, we get:
σ = (5.62 - (5.08 - Z1 * σ)) / Z2
Simplifying further, we get:
σ = (0.54 + Z1 * σ) / Z2
Solving this equation for σ by multiplying both sides by Z2:
Z2 * σ = 0.54 + Z1 * σ
(Z2 - Z1) * σ = 0.54
σ = 0.54 / (Z2 - Z1)
Substituting the values of Z1 and Z2 into the equation, we get:
σ = 0.54 / (-1.04 - (-0.84))
σ = 0.54 / (-0.20)
σ = -2.7 kg
Step 5: Calculate the mean μ.
Substituting the value of σ into equation 1, we get:
μ = 5.08 - (-0.84) * (-2.7)
μ = 5.08 + 2.25
μ = 7.33 kg
Therefore, the estimated mean of the masses of the boxes of apples is 7.33 kg, and the estimated standard deviation is 2.7 kg.
To estimate the mean and standard deviation of the masses, we can use the properties of the normal distribution. Let's denote the mean of the masses as μ and the standard deviation as σ.
First, we need to find the z-scores corresponding to the given percentages. The z-score represents the number of standard deviations a value is from the mean in a normal distribution.
For the first condition, where 20% of the boxes are heavier than 5.08 kg, we need to find the z-score corresponding to the 80th percentile. To find this, we can use a standard normal distribution table or a calculator.
The z-score corresponding to the 80th percentile is approximately 0.84. This means that the value 5.08 kg is 0.84 standard deviations below the mean.
Using the formula for the z-score:
z = (x - μ) / σ
we can rewrite this as:
0.84 = (5.08 - μ) / σ
Similarly, for the second condition, where 15% of the boxes are heavier than 5.62 kg, we need to find the z-score corresponding to the 85th percentile. The z-score corresponding to the 85th percentile is approximately 1.04. This means that the value 5.62 kg is 1.04 standard deviations below the mean.
Using the formula for the z-score once again:
1.04 = (5.62 - μ) / σ
Now we have a system of two equations with two unknowns (μ and σ). We can solve this system of equations to estimate the values of μ and σ.
First, we can rearrange the equations to solve for μ:
μ = 5.08 - 0.84σ
μ = 5.62 - 1.04σ
Setting these two equations equal to each other:
5.08 - 0.84σ = 5.62 - 1.04σ
Simplifying:
0.20σ = 0.54
Dividing both sides by 0.20:
σ = 0.54 / 0.20
σ = 2.70
Substituting this value back into either of the original equations:
μ = 5.08 - 0.84σ
μ = 5.08 - 0.84(2.70)
μ = 5.08 - 2.268
μ = 2.812
Therefore, the estimated mean (μ) of the masses is approximately 2.812 kg, and the estimated standard deviation (σ) is approximately 2.70 kg.