if the heights of men follow a normal distribution, and 99.7% fall between 5" and 7" what is the estimate standard deviation of the height?
Well, if 99.7% of men fall between 5" and 7" in height, it means that the remaining 0.3% must be extremely short or extremely tall individuals. And considering we're talking about height here, I highly doubt we're dealing with a population of circus performers!
But fear not, my friend, I have a delightful formula that will save the day. In a normal distribution, approximately 99.7% of values fall within three standard deviations from the mean. Since the range you provided is from 5" to 7", we can determine that the mean is at the half-point between these values, which is 6".
So, with three standard deviations encompassing 99.7% of the data, we divide the total range of 2" by 6 standard deviations (3 on each side) to get the approximate standard deviation. Thus, the estimate standard deviation of the height would be roughly 2" / 6 ≈ 0.33".
Of course, keep in mind that this is a simplified estimate, and real-world data may exhibit some variation. But who needs mathematical precision when you can just clown around with a good estimate? Enjoy!
To estimate the standard deviation of the height, we can use the rule for normal distributions, which states that approximately 99.7% of the values fall within three standard deviations of the mean.
In this case, we are given that 99.7% of the heights fall between 5" and 7". Since this range encompasses six standard deviations (from 5" to 6.99" is six standard deviations), we can calculate the estimate standard deviation using the formula:
Standard Deviation = (Range of data) / 6
Range = 7" - 5" = 2"
So, the estimate standard deviation of the height is:
Standard Deviation = 2" / 6 ≈ 0.333"
To find the estimated standard deviation of the height, we need to use the concept of the empirical rule, which states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
From the given information that 99.7% of the heights fall between 5" and 7", we can infer that this range represents three standard deviations from the mean.
Let's represent the mean of the height distribution as μ and the standard deviation as σ. Given that the range between 5" and 7" corresponds to three standard deviations, we have:
5σ ≤ 7σ
Dividing both sides by σ:
5 ≤ 7
This shows that the range of three standard deviations equals 2 inches. So, we can set up an equation:
3σ = 2
Dividing both sides by 3:
σ ≈ 0.67
Therefore, the estimated standard deviation of the height distribution is approximately 0.67 inches.
99.7% = mean ± 2.96SD
Assuming a normal distribution, the mean would be 6.
7 = mean + 2.96SD
Solve for SD.