- Suppose a variable X has a bell-shaped distribution with a mean of 150 and a standard deviation of 20.

a. What percentage of X values lies between 130 and 170?

b. What percentage of X values lies between 110 and 190?

c. What percentage of X values lies above 190? - Variable X has a mean of 15 and a standard deviation of 2.

a. What percentage of X values will lie within 1.5 standard deviation of the mean?

b. What is the minimum percentage of X values that lie between 8 and 17?

We do not provide complete solutions here. The purpose of homework is

that you learn things by doing them, and solving problems is something

questions like these try to teach.

If you’re just after precooked things you can put in an exam answer,

that is just cheating. I will not enable it.

Instead, how would *you* solve these problems? Find the definition of

these mathematical terms. Write in English (or whatever you usual

language is) how you might approach them if you were doing it by hand.

Then write little bits of code that do that for each piece.

Come here with questions, but bring your own code or specific questions

which prevent you trying to write code.

Cheers,

Cameron Simpson cs@cskk.id.au

import numpy as np

import math

import scipy.stats

import dill

x = scipy.stats.norm.pdf(130,150,20)

I am tried the above code, however, i am not sure.

This is a start. I’ve never used scipy myself, so we will learn

together.

The documentation for the scipy.stats package is here:

https://scipy.github.io/devdocs/reference/stats.html

Keep it to hand.

All of that said, I’m not sure you need scipy for this, although it

looks like it has functions to answer these questions. You might do

better to consult the Wikipedia article about the normal distribution:

and for what a standard deviation is:

The docs for `norm`

are here:

https://scipy.github.io/devdocs/reference/generated/scipy.stats.norm.html#scipy.stats.norm

and it looks like the `pdf`

function is a probably desnity function.

It isn’t clear to me that initialising a pdf with (130,150,20) is a

sensible thing to do. In particular, I’ve no idea what the first

parameter should be. The second parameter looks to be the location of

the mean, so 150 seems correct. The third parameter is means to be a

scale; it is possible that that this represents the size of the standard

deviation, but the docs are not clear to me.

It seems that the `interval(alpha,loc=0,scale=1)`

is a better way to

answer your percentage questions (right at the bottom of the

scipy.stats.norm.html page mentioned above). It takes:

```
interval(alpha, loc=0, scale=1)
```

where `alpha`

is a fraction of the distribution. So if `loc=150`

and

`scale=20`

you could try various values for `alpha`

between 0.0 and 1.0

until you get the endpoints you want, such as `130,170`

for your

question 1a.

Cheers,

Cameron Simpson cs@cskk.id.au

Which bell-shaped distribution?

It can make a very big difference between a normal (Gaussian)

distribution, a t-distribution, Cauchy distribution, etc.

But let’s use the normal distribution. There is no need for numpy or

scipy to solve this problem!

https://docs.python.org/3/library/statistics.html#normaldist-objects

```
>>> from statistics import NormalDist
>>> bell = NormalDist(mu=150, sigma=20)
>>> bell.cdf(170) - bell.cdf(130)
0.6826894921370859
```

That’s 68.3% of values are between 170 and 130, or another way of saying

this that is within one standard deviation of the mean.

(The mean, mu=150, plus or minus the stdev, sigma=20, is 170 and 130.)

```
>>> bell.cdf(190) - bell.cdf(170)
0.13590512198327787
```

13.6% of values are between 190 and 170.

```
>>> 1 - bell.cdf(190)
0.02275013194817921
```

2.3% of values are above 190.

I will leave Q2 to you.