The normal distribution & skewness (S1)

Selene · 14-05-2009: 14th May 2009 11:27

I'm having quite a bit of trouble getting the answers to some of the harder normal distribution questions in the textbook. I know how to standardise and draw diagrams, and thought I could tackle most questions, but maybe not!

It doesn't help that the S1 revision guide I ordered on the 2nd April hasn't come through yet, may have to cancel that one.

Anyway, I would really appreciate it if someone could run me through a question from the textbook and explain why each step is taken. I know that may look a bit stupid, but rather look stupid now than on results day!

Either one of the two below would be fantastic.

1) The thickness of some sheets of wood follows a normal distribution with the mean and standard deviation not given. 96% of the sheets will go through an 8mm gauge while only 1.7% will go through a 7mm gauge. Find the mean and standard deviation.

2) Records from a dental practice show that the probability of waiting to go into the surgery, for more than 20 minutes is 0.0239. If the waiting time is normally distributed with standard deviation 3.78 what is the mean waiting time?

Thank you so much for anyone who can help.

Also, which method of skewness would you use? Will it be enough to learn:
positive skew: mode < median < mean
negative skew mean < median < mode

or would you advise
positive q2-q1 < q3 - q2
symmetry =
negative >

Or something else?

darkness9999 · 14-05-2009: 14th May 2009 11:38

do u know the answers to Q 1? is it 0.26 and 7.55 ?

Jaziek · 14-05-2009: 14th May 2009 11:40

well for 1. have you got

P(T>8)= 4%
and
P(T<7)=1.7%

then standardize those, and you get 2 simultaneous equations which you can solve for mean and std. dev

Selene · 14-05-2009: 14th May 2009 11:44

Originally Posted by darkness9999

do u know the answers to Q 1? is it 0.26 and 7.55 ?

Yep!

Could you explain how you got there?

Selene · 14-05-2009: 14th May 2009 11:45

Originally Posted by Jaziek

well for 1. have you got

P(T>8)= 4%
and
P(T<7)=1.7%

then standardize those, and you get 2 simultaneous equations which you can solve for mean and std. dev

This is what I did, but didn't get the right answer. Probably just some silly working mistake somewhere. I'll try it again.

Thank you!

darkness9999 · 14-05-2009: 14th May 2009 11:49

Originally Posted by Selene

Yep!

Could you explain how you got there?

Q1
$P(Z\leq \frac{7-\mu}{\sigma})=0.017 \Rightarrow P(Z\leq \frac{\mu -7}{\sigma})=0.9830 \Rightarrow \frac{\mu -7}{\sigma}= 2.12 P(Z\leq \frac{8-\mu}{\sigma})=0.96 \Rightarrow \frac{8 -\mu}{\sigma}= 1.75$
So now you've got 2 equations with two unknowns ... u can solve them simoultaneously

darkness9999 · 14-05-2009: 14th May 2009 11:54

Q2

$P(Z\geq \frac{20-\mu}{3.78})= 0.0239$

$\Rightarrow P(Z\leq \frac{20-\mu}{3.78})= 0.9761$

$\Rightarrow \frac{20 -\mu}{3.78}= 1.98$

$\Rightarrow \mu= 12.52 (4sf)$

I hope its all clear to u

Selene · 14-05-2009: 14th May 2009 11:57

Originally Posted by darkness9999

Q2

$P(Z\geq \frac{20-\mu}{3.78})= 0.0239$

$\Rightarrow P(Z\leq \frac{20-\mu}{3.78})= 0.9761$

$\Rightarrow \frac{20 -\mu}{3.78}= 1.98$

$\Rightarrow \mu= 12.52 (4sf)$

I hope its all clear to u

Thanks, the more help I get on this the better! I would rep you but I don't think I'm worth anything!

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The normal distribution & skewness (S1)