in a distribution of 160 values with a mean of 72,at least 120 fall within the interval 67-77. approximately what percentage of values should fall in the interval 62-82? use chebyshev's formula.
S= 2.5
S= 2
To approximate the percentage of values that fall in the interval 62-82 using Chebyshev's inequality, we need to find the range of values within two standard deviations from the mean.
Chebyshev's inequality states that for any distribution, the proportion of values within k standard deviations of the mean is at least 1 - 1/k^2, where k is any number greater than 1.
In this case, we will use k = 2 standard deviations.
First, we need to find the standard deviation (σ) for the distribution. We can use the following formula:
σ = √(variance)
Since we don't have the variance directly given, we need to find it using the information provided.
The variance (variance) is calculated as the square of the standard deviation:
variance = σ^2
For Chebyshev's inequality, we are given that at least 120 values fall within the interval 67-77. We can use this information to find the minimum value for variance:
P(67 ≤ X ≤ 77) ≥ 120/160
P(67 ≤ X ≤ 77) ≥ 0.75
To calculate the minimum variance, we assume the worst-case scenario, where the probability is at its minimum value. In this case, we have:
P(67 ≤ X ≤ 77) = 1 - P(X ≤ 67) - P(X ≥ 77)
Since the distribution is symmetric, P(X ≤ 67) is the same as P(X ≥ 77), and let's denote it as p:
0.75 = 1 - 2p
2p = 0.25
p = 0.125
So, the minimum variance is:
variance = (10^2) / ((2 * 0.125)^2)
variance = 1600
Now that we have the variance, we can find the standard deviation:
standard deviation (σ) = √1600 = 40
Next, we can calculate the range of values within two standard deviations from the mean:
Lower Limit = mean - (2 * σ) = 72 - (2 * 40) = 72 - 80 = -8
Upper Limit = mean + (2 * σ) = 72 + (2 * 40) = 72 + 80 = 152
So, the range of values within two standard deviations from the mean is -8 to 152.
To find the approximate percentage of values that fall in the interval 62-82, we need to calculate the proportion of values within this range compared to the range between -8 and 152.
Percentage = (82 - 62) / (152 - (-8)) * 100
Percentage = 20 / 160 * 100
Percentage = 12.5%
Therefore, approximately 12.5% of the values should fall in the interval 62-82 using Chebyshev's formula.
To approximate the percentage of values that should fall in the interval 62-82 using Chebyshev's inequality, we can follow these steps:
Step 1: Calculate the standard deviation (σ) of the distribution.
To use Chebyshev's inequality, we need to know the standard deviation. Unfortunately, the question does not provide the standard deviation. However, we can use another piece of information to estimate it.
Step 2: Use the given information to estimate the standard deviation.
Since the interval 67-77 contains at least 120 values, we can assume that it covers a significant portion of the distribution. Let's assume that 120 values fall within the range of 67-77.
We can estimate the standard deviation (σ) using the formula:
σ = (range / 6) * sqrt(n)
where range is the width of the interval (77 - 67 = 10), and n is the number of values falling within that interval (120).
σ = (10 / 6) * sqrt(120) ≈ 8.16
Note that this is just an estimate and a rough approximation. The actual standard deviation might differ.
Step 3: Apply Chebyshev's inequality formula.
Chebyshev's inequality states that for any interval [mean - k * σ, mean + k * σ], where k is any positive number greater than 0, at least (1 - 1/k^2) of the values will fall within that interval.
In our case, the interval is [62, 82] and the mean is 72. The standard deviation (σ) estimated in Step 2 is approximately 8.16.
Using Chebyshev's inequality formula, we can calculate the percentage of values that should fall within the interval 62-82:
P(62 ≤ X ≤ 82) ≥ 1 - 1/k^2
In this case, we can assume k as a large value like 3 or 4 to guarantee a conservative estimate.
Using k = 4:
P(62 ≤ X ≤ 82) ≥ 1 - 1/4^2
P(62 ≤ X ≤ 82) ≥ 1 - 1/16
P(62 ≤ X ≤ 82) ≥ 15/16
So, at least 15/16, or 93.75%, of the values should fall within the interval 62-82, according to Chebyshev's inequality.