How do you solve these questions?

Question

Hello! I would like to ask on how to solve these questions in Statistics. The topic is about the normal distribution curve. (Before starting, I do not have a picture of the z table to show you what are the certain numbers to find to be able to answer the questions.. thus, a BRIEF and VIVID steps to be able to answer it is what I am asking for. To add more, how do you interpret each letters of the numbered questions? I'm finding it hard to interpret them.)

 

These are the questions:

1. Given normal distribution with µ=40 and σ=6

    a. Find the probability that X assumes a value greater than 50.

    b. The value X that has 38% of the area below it.

 

2. If an adult male cholesterol is normally distributed with µ=200 and σ=25, what is the probability of selecting a male whose cholesterol is

    a. Less than 165

 

3. If the weight of 100 students of a certain school closely follow a normal distribution with mean of 50.0 kgs and a standard deviation of 1.2 kgs.

    a. What is the probability that a student picked at random weighs less than 53.0 kgs?

    b. At least what weight should a student have to belong to the upper 17%?

    c. How many are heavier than 49.2 kgs?

 

That is all guys! Thank you so much for taking your time to read my long questions. But I REALLY need your help on how to solve this one cuz' it's been driving me insane since yesterday. I tried several ways to get the answers but to no avail. That's why, this site is my only choice to help me solve these questions in Statistics. I hope you can understand :((

Answer 1

Good morning, Paige!

 

Sounds like these have been bugging you.  :)

 

How about a few hints to see if we can get you on the right track for the first question, and maybe that'll get you on your way for the rest, too.

 

1.  You mentioned that you don't have a picture of the table for the z-statistic to show us.  However, I'm assuming that you actually have it.  Otherwise, these questions are pretty tough!  As in, they're impossible.  :)

 

2.  So, what in the heck is a z-score?  If I ask you about a variable that is normally distributed, with μ=0 and σ=1, and if I ask you what the probability is that z < 0.600, would you be able to do that, using the table for the standard normal distribution?

 

3.  How about if I ask you how to figure out the probability that z > 0.600 (greater than instead of less than)?  That is, do you know how to use the standard normal distribution table to figure out probabilities that aren't quite listed, but you can add/subtract values to get there?

 

Up until that point, it's all about the standard normal distribution.  Z gives us a way of talking about variables that are normally distributed, on a specific standard scale.  For Z, the mean is 0, and the standard deviation is 1.  The "normal" part tells us that we're dealing with a bell curve, and the "standard" part tells you where the curve is (the mean) and how curved it is (the standard deviation).

 

Are ya with me so far?  If not, then we need to back up further and make sure we get the concept of a normal distribution and the standard normal distribution.  For the moment, though, I'll assume that we're ok.

 

4.  So, let's talk about Question #1.  We have a normal distribution, and we are given a mean and a standard deviation.  In the problem, we're talking about a variable that has a mean of 40 and a standard deviation of 6. Maybe a bunch of students took a test with 60 questions, and they averaged 40 correct with a standard deviation of 6. And now, we want to know what percentage of the students got more than 50 correct. That's the "X > 50" part.  Uh oh, but there's a problem.  We can't use the standard normal distribution table to calculate any probabilities!  Why not?

 

5.  If you can answer Question #4, then we should now be asking ourselves "how in the heck do I turn this problem into something that looks like the standard normal distribution"?  I'm looking at a score of 50 (X>50), but that's relative to a mean of 40 and a standard deviation of 6.  z is relative to a mean of 0 and a standard deviation of 1. 

 

You've got a formula in your textbook somewhere that tells you how to do the conversion.  But let's talk about the concept here. 

 

- In this problem, let's take a slightly different example.  If I were asking about X > 40 instead of X > 50, then I'm asking about a score that is greater than the mean (because the mean is 40).  On the standard normal distribution, that's like asking if z is bigger than the mean...and what's the mean of the standard normal distribution? Zero.  Your mean is 40, but the standard normal mean is 0, so as a "z," your X > 40 is the same as z > 0.  So, that conversion we can sorta do by hand.

- In this problem, with a standard deviation of 6, then 46 would be exactly one standard deviation away from the mean, right?  Take 40, add 6, and I get to 46.  So, if I asked you what the chances are that X > 46, what I'm asking is the same as "what are the chances that X is more than one standard deviation away from the mean"?  Let's ask that same question for the standard normal distribution.  On that scale, the standard deviation is 1.  So, for X > 46 in our specific problem, that's like saying that z > 1 on the standard normal distribution.  

 

Those two are a little more straightforward, because I picked values of X that sit exactly at the mean, or exactly on a standard deviation.  But that doesn't always happen. Let's go back to the problem, which wants you to figure out the probability that X > 50.  That's, like, more than one standard deviation away from the mean of 40...but exactly how many?  If we can figure that out, then we can get a z score, and then we can use the standard table to get the answer.

 

If you have the formula for converting an X score to a z score, you should be on your way.

 

Let's start there.  Hope this helps get you going,

 

-- Michael