Question

For a normal distribution, it is known that the mean is zero and the standard deviation is 1. Be sure to draw and label a normal distribution for each problem. Find the probability that a z score is between -1.15 and 1.98.

Answer 1

To find the probability between two z-scores on a normal distribution, we can use a z-score table.

A z-score table gives the probability of a data less than the corresponding z-score, like in the picture:

So, if we want the probability between 1.98 and -1.15, we need to do the following:

$P(-1.15However, usually z-score tables go only from z = 0 and above, so we need a way to consult the negative z.

Since the normal distribution in symmetric around z = 0, we have that:

[tex]P(z<-1.15)=P(z>1.15)$

And since the whole distribution give a 100% probability, or an area of 1, we have that:

$\begin{gathered} P(z<1.15)+P(z>1.15)=1 \\ P(z>1.15)=1-P\mleft(z<1.15\mright) \end{gathered}$

So, in the end, we have:

$\begin{gathered} P(-1.151.15) \\ P(-1.15So, we can consult the values for the probabilities of z < 1.98 and z < 1.15 on the z-score table:[tex]\begin{gathered} P(z<1.98)\approx0.9761 \\ P(z<1.15)\approx0.8749 \end{gathered}$

So, the total probability is:

[tex]\begin{gathered} P(-1.15So, the probability is approximately 0.8510.