1. ## Normal distribution

Hi,

I need help with the setup of this problem.

The capacity for the elevator is 2500 kgs. Baseball players have weights that are normally distributed with a mean of 120 kgs and a standard dev. of 80 kgs. 25 baseball players get on the elevator. What is the likelihood that the cable will snap?

I set it up as Z= (X-µ) / σ but what is X ?

2. Originally Posted by depaulie10
Hi,

I need help with the setup of this problem.

The capacity for the elevator is 2500 kgs. Baseball players have weights that are normally distributed with a mean of 120 kgs and a standard dev. of 80 kgs. 25 baseball players get on the elevator. What is the likelihood that the cable will snap?

I set it up as Z= (X-µ) / σ but what is X ?
X=100

You could say that the average weight of 25 players is 25(120)
and the standard deviation of weight for 25 players is 25(80) kg.

then

$\displaystyle \frac{X-\mu}{\sigma}=\frac{25(100)-25(120)}{25(80)}=\frac{100-120}{80}$

which is similar to dealing with a smaller elevator of capacity 100kg for a single player.

3. Thanks for your help. I understand that part so now am I just supposed to look up the Z-score? What do I look for then?

4. Originally Posted by depaulie10
Thanks for your help. I understand that part so now am I just supposed to look up the Z-score? What do I look for then?
You may be using Z-values that start at zero or a table of positive and negative z-values,

as clearly in this case z is negative.

In this example you are calculating the probability that Z is greater that your z-score.

Hence, if you only have positive values of z, you need to find the probability
that Z is less than the modulus of your z-score.

Since the average weight of the 25 players exceeds the breaking strain of the cable,
there is a more than 50 percent chance the cable will snap.

You could also calculate $\displaystyle z=\frac{120-100}{80}=0.25$

and look up the probability that $\displaystyle Z\ \le\ 0.25$

Z tables give you the probability of z being less than or equal to some z-score. the normal distribution is symmetrical about the mean,
so there is flexibility in making calculations.

5. Thanks so much. I emailed my professor regarding this problem but he was not helpful at all. It'll be a long semester but I appreciate the help!

6. Originally Posted by depaulie10
Hi,

I need help with the setup of this problem.

The capacity for the elevator is 2500 kgs. Baseball players have weights that are normally distributed with a mean of 120 kgs and a standard dev. of 80 kgs. 25 baseball players get on the elevator. What is the likelihood that the cable will snap?

I set it up as Z= (X-µ) / σ but what is X ?
The random variable is $\displaystyle W = X_1 + X_2 + .... + X_{25}$ where $\displaystyle X_i$ ~ Normal $\displaystyle (\mu = 120, \, \sigma = 80)$.

You're expected to know that the sum of normal random variables is also a normal random variable: Sum of normally distributed random variables - Wikipedia, the free encyclopedia.

You should therefore know that in this case, W ~ Normal $\displaystyle (\mu = 25 \times 120 = ...., \, \sigma = \sqrt{25 \times 80^2} = ....)$.

Use the distribution of W to calculate Pr(W > 2500).

7. Originally Posted by mr fantastic
The random variable is $\displaystyle W = X_1 + X_2 + .... + X_{25}$ where $\displaystyle X_i$ ~ Normal $\displaystyle (\mu = 120, \, \sigma = 80)$.

You're expected to know that the sum of normal random variables is also a normal random variable: Sum of normally distributed random variables - Wikipedia, the free encyclopedia.

You should therefore know that in this case, W ~ Normal $\displaystyle (\mu = 25 \times 120 = ...., \, \sigma = \sqrt{25 \times 80^2} = ....)$.

Use the distribution of W to calculate Pr(W > 2500).
Ah, well now I'm confused because I don't recall anything about a random variable W in lecture. Is the above explanation by Archie not right then?

8. Originally Posted by depaulie10
Ah, well now I'm confused because I don't recall anything about a random variable W in lecture. Is the above explanation by Archie not right then?
W is just the name I have given the random variable defined by the sum of the other random variabls. If you don't like W, use Y or U or Fred. I would not have bothered to make my reply if I thought that the replies you had already got were correct.

9. Well thanks. So how do I go on calculate Pr(W > 2500)...

10. Originally Posted by depaulie10
Well thanks. So how do I go on calculate Pr(W > 2500)...
Haven't you been taught how to calculate probabilities from a normal distribution? What do your classnotes and textbook say about getting the z-value, using tables etc. (Is this the very first question about calculating probability from a normal distribution that you have ever tried?)

11. I took one statistics class 3 years ago. The class is econometrics and I have 8 pages of class notes that are not helpful and contain little in the form of valuable examples which is why I'm on here. I've spent hours on this homework trying to understand and at this point I'm just trying to work my way backwards since the professor has not responded to my emails either. I plan on going to a tutor on Monday but I want to see what it is that I need help with! Thanks for responding anyway.

12. Originally Posted by depaulie10
I took one statistics class 3 years ago. The class is econometrics and I have 8 pages of class notes that are not helpful and contain little in the form of valuable examples which is why I'm on here. I've spent hours on this homework trying to understand and at this point I'm just trying to work my way backwards since the professor has not responded to my emails either. I plan on going to a tutor on Monday but I want to see what it is that I need help with! Thanks for responding anyway.
Step 1: $\displaystyle Z = \frac{X - \mu}{\sigma}$ or, since the normal variable is W in this case, $\displaystyle Z = \frac{W - \mu}{\sigma}$.

Substitute the mean and the standard deviation of W into the above to calculate the z-value.

Do that and then I will go to step 2.

13. Thanks.

Z= W - 3000 / 400

14. Originally Posted by depaulie10
Thanks.

Z= W - 3000 / 400
You also need to substitute W = 2500. Therefore Z = -1.25.

Step 2: Calculate Pr(Z > -1.25).

Note that Pr(Z > -1.25) = Pr(Z < 1.25) using the symmetry of the normal distribution. Pr(Z < 1.25) is something you can look up in your standard normal distribution tables ....

15. Which is 0.8944.

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