Suppose X is a random variable with mean μ and standard deviation σ. If a large number of trials is observed, at least what percentage of these values is expected to lie between μ − 3σ and μ + 3σ? (Round your answer to the nearest whole number.)
Look in the back of your statistics textbook for a table called something like “area under normal distribution” to find the proportion/probability of Z = ±3.
Or you can use this:
http://davidmlane.com/hyperstat/z_table.html
To determine the percentage of values expected to lie between μ - 3σ and μ + 3σ, we can refer to the concept of the empirical rule, also known as the 68-95-99.7 rule.
According to this rule, for a normally distributed random variable:
- Approximately 68% of the values fall within 1 standard deviation of the mean.
- Approximately 95% of the values fall within 2 standard deviations of the mean.
- Approximately 99.7% of the values fall within 3 standard deviations of the mean.
Since we are interested in the range μ - 3σ to μ + 3σ, which is three standard deviations on each side of the mean, we can conclude that approximately 99.7% of the values are expected to lie within this interval.
Therefore, the answer to the question is 99%. (Rounding to the nearest whole number).