Question

Suppose your manager indicates that for a normally distributed data set you are analyzing, your company wants data points between

z
=
−
1.5
and
z
=
1.5
standard deviations of the mean (or within 1.5 standard deviations of the mean). What percent of the data points will fall in that range?

Answer 1

For data points between z = -1.5 and z = 1.5 standard deviations from the mean in a normally distributed data set, approximately 86.6% of the data falls within that range, as determined by consulting a Z-Table.

Step-by-step explanation:

If you are analyzing a normally distributed data set and are looking for data points between z = -1.5 and z = 1.5 standard deviations of the mean, you can look to the empirical rule, also known as the 68-95-99.7 rule, to help find your answer. This rule states that approximately 68% of data falls within one standard deviation, about 95% within two, and about 99.7% within three standard deviations of the mean. To find the percentage that falls within 1.5 standard deviations, you would have to use a Z-Table of Standard Normal Distribution or other statistical tools.

Generally, a range of data between z-scores of -1.5 and 1.5 would capture more than 68% but less than 95% of data points. As a rough estimate, since 95% of data is within two standard deviations, and the distribution is symmetric, you might deduce that approximately 90% of the data falls within 1.5 standard deviations, since it is halfway between one and two standard deviations. However, for a precise calculation, the aforementioned Z-Table is necessary. The exact percentage for z-scores between -1.5 and +1.5 is approximately 86.6%, by looking up the values in a standard normal distribution table.

Answer 2

86.64% of the data points will fall in that range

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem:

z = -1.5 has a pvalue of 0.0668

z = 1.5 has a pvalue of 0.9332

0.9332 - 0.0668 = 0.8664

86.64% of the data points will fall in that range