Question

What percentage of the population values fall under $1.5$ standard deviations in a normal distribution?

Answer 1

Since the given population value is $1.5$, consider the area that falls under$-1.5$ and $1.5$. Upon calculating the probabilities, we get the required percentage. We can determine the population when it is normally distributed as the population has a precisely normal distribution if the mean, median and mode have equal values.

Complete step by step solution:
Now let us learn about the population in a normal distribution. The normal distributions are bell-shaped and symmetric. Most of the members of the normally distributed population have values close to mean. We can say if the data is normally distributed or not by checking normality by plotting a frequency distribution. The steeper the bell curve, the smaller the standard deviation.
Now let us start solving our problem.
Firstly, consider the area that falls under $-1.5$ and $1.5$.
We get,

& \Rightarrow P\left( Z<\left| 1.5 \right| \right) \\\ & =P\left( -1.5 < Z < 1.5 \right) \\\ & =P\left( Z < 1.5 \right)-P\left( Z < -1.5 \right) \\\ & =1-P\left( Z < -1.5 \right)-P\left( Z < -1.5 \right) \\\ & =1-2\times P\left( Z<-1.5 \right) \\\ \end{aligned}$$ Upon solving this, we get $$\begin{aligned} & P\left( Z<\left| 1.5 \right| \right)=1-2\times 0.0668 \\\ & =0.8664 \\\ \end{aligned}$$ Since, we are asked to calculate the percentage, let us convert the obtained decimal into the percentage by multiplying it with hundred. On multiplying, we get the percentage as $$\begin{aligned} & \Rightarrow 0.8664\times 100 \\\ & 86.6\% \\\ \end{aligned}$$ **$$\therefore $$ The percentage of the population values fall under $$1.5$$ standard deviations in a normal distribution is $$86.6\%$$** **Note:** If our data is not normally distributed, then we should not assume normality. Generally, for large size population samples, it is normally distributed. The characteristics of a normal distribution is that it is symmetric, uni modal and asymptotic . A normal distribution is perfectly symmetrical around its centre.