Clarification about normal distribution issues I'm doing the following self-help revision question on Normal distribution. I don't quite seem to understand: 


*

*the answer in Part II and how it is derived

*for Part III, why is is that p = 0.1075 

*for Part III, why does  P( Y = 5 ) not have a multiplication with (1 - 0.1075) like P( Y = 4 )


Appreciate some advice and directions please.
Question

Answer Key

 A: 
the answer in Part II and how it is derived

The answer scheme decomposes the question into two separate tasks. First, it ignore the lower end and just look at where should one cut a normal distribution so that 25% is cut off from the right side.
Then, we repeat by now ignoring the upper end, and just look at where should one cut a normal distribution so that 25% is cut off from the left side.
Now, we know two number that can cut off the upper and the lower 25%, what left would be the middle 50%, which is what the question asks for.

for Part III, why is is that p = 0.1075

This is relevant to these parts that I circled:

If you understand part i (on which you didn't ask any question so I assumed), you should understand why 0.1075.

for Part III, why does P(Y = 5) not have a multiplication with (1 -
  0.1075) like P(Y = 4)

Because when there are only 4 of them with the same trait, you'll need to factor in the probability drawing the remaining 5th bag that does not have the trait. Seeing that term as $(1-0.1075)^1$ may help you understand, but in math notation, we simplify by taking that power to 1 out.
When all five have the trait, you can still multiple that term, but the term would be $(1-0.1075)^0$, which is 1 so there is no need to explicitly show it.


Thanks Penguine. Really appreciate it. However may i ask where does
  the 0.6745 in part II come from?

That is the z-score that corresponds to cutting the normal distribution at the 75% point. Likewise, the negative one is the point where you can cut at the 25% point.


These two screenshots are from an online calculator showing the same z-scores.


Thanks penguin. Thats perfect! One more thing - how does the equation
  translate to b = 5 + 1.25 + [Penguin Knight, should be x, not +] 0.6745

Let's just say the unit of the whole formula is in "pound per square inch." We know that the mean is 5. And we know that the standard deviation is 1.25. One standard deviation is 1 z-score. If you want 0.67 SD more on top of the mean, how would you go getting it?
