ASAP without being rounded please

68% (One Standard Deviation): In a normal distribution (bell curve), approximately 68% of the data falls within one standard deviation from the mean. This means that if you have a set of data with a normal distribution, about 68% of the values will be within this range.

95% (Two Standard Deviations): About 95% of the data falls within two standard deviations from the mean. In other words, if your data follows a normal distribution, roughly 95% of the values will lie within this wider range.

99.7% (Three Standard Deviations): This represents an even wider range. About 99.7% of the data falls within three standard deviations from the mean. It encompasses almost all the data points in a normal distribution.

To find number of standard deviations from mean for any given value X, the formula is:
[tex]z = \dfrac{X - \mu}{\sigma}\\[/tex]

where
X is the value of interest
[tex]\mu[/tex] is the mean
[tex]\sigma[/tex] is the standard deviation
z is the standard normal score which indicates the number of standard deviations from the mean

Once we get z, we can look at the standard normal tables (or use a calculator to find different probabilities)

But in this case, we really don't need to go through this exercise since the graph itself provides the values at 1, and 2 standard deviations from the mean

We are asked to find P(20 < v < 50)

P(20 < v < 50) = P(20 < 40) + P(40 < 50) where 40 is the mean

P(20 < 40) represents the area between 20 and 40 on the normal distribution curve

P(40<50) represents the area between 40 and 50 on the normal distribution curve
We know that 95% of the values lie between -2 and +2 standard deviations. This means that the area between 20 (40 - 20) and 60(40 + 20) should be 95%
Since the distribution is symmetrical, we can state that P(20 < 40) = 95%/2 = 47.5%
Similarly we see that 50 lies 1 standard deviation from 40 and P(40 < 50) = 68%/2 = 34%
So P(20 < v < 50) = P(20 < 40) + P(40 < 50)
= 47.5% + 34%
= 81.5%