For the proportion of data that falls between two values in a standard normal distribution.
In such a task, we need to find the cumulative distribution function (CDF) values corresponding to our lower and upper bounds. The CDF gives us the proportion of data that falls below a given point.
Let's assume that the values of Plu and O are -1 and 1, respectively. It is an assumption, since, unfortunately, you haven't indicated the particular values in the task.
Step 1: Find the proportion below the lower bound (-1)
Using the cumulative distribution function, the proportion of values below -1 in a standard normal distribution is about 0.1587.
Step 2: Find the proportion below the upper bound (1)
Similarly, the proportion of values below 1 in a standard normal distribution is about 0.8413.
Normally, we would stop here if we were just looking for the proportion below a certain point. However, you're looking for the proportion between two values, so there's an additional step.
Step 3: Subtract to find the proportion between the lower and upper bounds
The proportion of values between -1 and 1 is equal to the proportion of values below 1, minus the proportion of values below -1.
Mathematically, that's 0.8413 - 0.1587 = 0.6827.
So, the proportion of data that falls between -1 and 1 in a standard normal distribution is about 0.6827, or 68.27%.
Keep in mind that standard normal distribution values range from negative infinity to positive infinity with 0 as the mean. So, when we say the proportion of data between -1 and 1, we're actually referring to the proportion of data within one standard deviation of the mean.
And as a final note, provide the specifics about the Plu and O values for a more precise calculation.