How Do You Calculate y1 and y2 for a Given Probability in a Normal Distribution?

[someguy54]

Need a little help here:

Find the random variable coefficients y1 and y2 where P(y1 < y < y2) = 0.5. Where mean is 0.7 and standard deviation is 0.03 (not sure if you need that). I have no clue where to start with this one.

Thanks for any help

 

[HallsofIvy]

Perhaps because there are an infinite number of answers! [itex]-\infty to 0.7[/itex] would obviously work because of the symmetry of the normal distribution about the mean. So would [itex]0.7 to \infty[/itex]. For finite values of y1 and y2, try this. Convert to the "standard" z-score using [itex]z= (y- \mu)/\sigma[/itex] which here is [itex]z= (y- 0.7)/0.03[/itex]. Pick any y1 you want, less than the mean, and calculate its z-score. [For example, choosing (just because it makes the calculation easy) y1 to be 0.67, we get z= -0.03/0.03= -1]. Look that up on a table of the normal distribution (a good one is at http://people.hofstra.edu/Stefan_Waner/RealWorld/normaltable.html ) to find P(y1) [for z= -1 I get 0.46587] If that is less than 0.5, add it to 0.5 to see how much "more" you need and look up the z corresponding to that and, finally, compute the y2 that gives. [0.46587+ 0.5= 0.96587. The table says that corresponds to z= 1.82 and then 1.82= (y2- 0.7)/0.03 gives y2= 0.7546. You can choose any y1 you want, less than 0.7, and do the same to get a different y2.

 

[tacman]

!

To solve this problem, we need to use the properties of the normal distribution. The normal distribution is a symmetrical bell-shaped curve that is characterized by its mean and standard deviation. The mean represents the center of the distribution, while the standard deviation measures the spread or variability of the data.

In this case, we are given the mean (0.7) and standard deviation (0.03) and we need to find the values of y1 and y2 such that the probability of a random variable falling between them is 0.5. This means that the area under the curve between y1 and y2 is equal to 0.5.

To solve for y1 and y2, we can use a table of standard normal probabilities or a calculator. For simplicity, let's use a calculator.

Step 1: Convert the given mean and standard deviation to z-scores.

A z-score represents the number of standard deviations a data point is from the mean. We can calculate the z-score using the formula: z = (x - mean) / standard deviation.

For y1, we have z1 = (y1 - 0.7) / 0.03
For y2, we have z2 = (y2 - 0.7) / 0.03

Step 2: Find the z-scores that correspond to a cumulative probability of 0.25 and 0.75.

Using a standard normal distribution table or a calculator, we can find that the z-score for a cumulative probability of 0.25 is -0.6745 and the z-score for a cumulative probability of 0.75 is 0.6745.

Step 3: Substitute the z-scores into the equations from step 1 and solve for y1 and y2.

For y1, we have -0.6745 = (y1 - 0.7) / 0.03
Solving for y1, we get y1 = 0.6819

For y2, we have 0.6745 = (y2 - 0.7) / 0.03
Solving for y2, we get y2 = 0.7181

Therefore, the values of y1 and y2 that satisfy the given condition are 0.6819 and 0.7181, respectively.

In summary, to find the random variable coefficients y1 and y