In Excel we will use the command/function NORM. INV to get the inverse cdf of normal distributions. This command takes a left tail area (between 0 and 1) as it's first argument. The weights of bags of cookies are normally distributed with a mean of 15 ounces and a standard deviation of 0.85 ounces. Bags of cookies that have weights in the upper 7.5% are too heavy and must be repackaged. What is the most a bag of cookies can weigh and not need to be repackaged?

And if we solve for a we got

So the value of height that separates the bottom 92.5% of data from the top 7.5% is 16.223.

And using the NORM.INV function in excel we have:

"=NORM.INV(1-0.075,15,0.85)"

Snd we got the same result

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:

Where and

For this part we want to find a value a, such that we satisfy this condition:

  (a)

  (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.925 of the area on the left and 0.075 of the area on the right it's z=1.439. On this case P(Z<1.439)=0.925 and P(z>1.439)=0.075

If we use condition (b) from previous we have this:

But we know which value of z satisfy the previous equation so then we can do this:

And if we solve for a we got

So the value of height that separates the bottom 92.5% of data from the top 7.5% is 16.223.

And using the NORM.INV function in excel we have:

"=NORM.INV(1-0.075,15,0.85)"

Snd we got the same result

step-by-step explanation:

the shape,the size and the distance