a) If you have scores and you don't know if they are normally distributed, find the minimum percentage of scores that fall within 2.2 standard deviations on both sides of the mean? b) If you have scores that are normally distributed, find the percentage of scores that fall within 2.2 standard deviations on both sides of the mean? c) If you have scores and you don't know if they are normally distributed, how many standard deviations on both sides of the mean do we need to go to have 64 percent of the scores?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**Understanding Chebyshev's Formula for Data Distribution**
For data that is not normally distributed, we can't use z-scores. However, there is an equation that works on any distribution. It’s called Chebyshev’s formula. The formula is:
\[ p = 1 - \frac{1}{k^2} \]
where \( p \) is the minimum percentage of scores that fall within \( k \) standard deviations on both sides of the mean. Use this formula to answer the following questions.
**a)** If you have scores and you don't know if they are normally distributed, find the minimum percentage of scores that fall within 2.2 standard deviations on both sides of the mean: [______]
**b)** If you have scores that are normally distributed, find the percentage of scores that fall within 2.2 standard deviations on both sides of the mean: [______]
**c)** If you have scores and you don't know if they are normally distributed, how many standard deviations on both sides of the mean do we need to go to have 64 percent of the scores? [______]
*Note:* To answer part c, you will need to solve the equation for \( k \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F020b4ff6-36e2-475f-88e0-c8be863ccb7d%2Ff392cdba-6ada-48b3-9f91-c232cb022c58%2Fntvptif_processed.jpeg&w=3840&q=75)
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