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hi everybody

this is eugene lachlan and welcome to my

series of short how-to videos

in this video we're going to learn how

to calculate the confidence interval for

a sample and we're going to learn how to

do this by hand

so first off let's take a look at the

data that we're going to use in this

example

um i've used this data before in

previous videos these are the data for

14 randomly selected bags of food which

weigh one kilogram or 1 000 grams so

we've randomly selected those

from a production line as part of a

quality control process

and we want to be able to do do some

statistical calculations

with these data so the first thing i've

done with these

sample of 14 here is i've worked out the

sample mean and we can see the sample

mean x bar is equal to 996.21 grams

s is the standard deviation that works

out at 5.65

and n the sample size is 14. so we're

going to need these some of these values

in our later calculations

let's take a look at the t distribution

this is a typical t distribution our

friend the bell shape diagram here and

what we say for 95 percent confidence is

that um if it's bell-shaped like this

then with the

population mean mu in the center here 95

percent of all values

fall between the this area here the vast

majority of the bell

shape and five percent fall uh in

into the left and the right tail so half

of five percent in the left tail and

half of right five percent in the right

tail

so what we need to know here for our

sample size is what is the value what is

the t

value here uh at each of these and

margins at each of the tails so we're

going to need to look those up

in t tables so let's go ahead and do

that first

these are t tables here and they're

widely available online

and there's two things we need to know

we need to know down the left-hand side

here

what the row is and that's based on df

degrees of freedom so we're going to

need to have to work out the degrees of

freedom for our sample size

of 14 and we also then need to know what

what is the confidence level

that we want to do our test at so we're

doing it at 95 percent here

or a value of 0.05 so the degrees of

freedom first of all are straightforward

enough

and the degrees of freedom is equal to n

minus 1 which is equal to in our case 14

minus 1

which is equal to 13. so we have 13

degrees of freedom so let me go and look

this up on the table

so my row here of 13 so somewhere on

this row

if i just highlight the row somewhere on

this row

is my t value now how do i select the

column

well the column is i can do it in two

ways uh one is i can cross the bottom in

this version of a t table look and see

uh 95 here so that'll be in this column

or at the top

i can choose we've got two tails the

value of 0.05

so so somewhere on this column

is going to be my t value and where the

two values cross over

is the t value i'm going to need we can

see that that is 2.160

by the way if you wanted to do a 99 or a

different

value here you would have the same

degrees of freedom in this sample size

but you can see here i would move over

so for example at 99

confidence interval my t value is going

to be 3.012

but we're going to stick with 95

confidence for the moment

so my t value my t

value is equal to

2.160 so that's these values here is

2.160 and minus 2.160 here in the left

tail

now that value is going to become

important in a moment

next we need to be able to work out the

standard error so the standard error

formula our standard error of the mean

is a straightforward formula we're using

the sample standard deviation here

s and divided it by the square root of n

which in our case is 14. so we have the

two values here so let's go ahead and

work this out

so that is equal to a standard deviation

of 5.65

divided by the square root of 14. let me

get my calculator out and work that out

so 5.65 divided by

putting 14 and use the square root

symbol

to get that and then press equals and

that will give me a standard error of

1.51 i'll use two decimal places there

so now we've got the standard error and

our next job then is

to multiply our standard error so we

want to multiply our standard error

multiplied by

our t value so that is equal to

1.51 from here multiplied by

2.160

so let's work that out so we have the

1.51 still in our calculator

multiply that by 2.16

that is equal to 3.8

so now we have the value that we need to

estimate our upper value and our lower

value for the confidence interval

so let me put in the upper one first

and that's going to be equal to my mean

996.21 plus this value that i've just

worked out here

so that's going to be 996.21

plus 3.26 and that's going to be equal

to let me work that out

my calculator so 996.21

plus 3.26 is equal to

so that's 99.99999

and my lower limit for the confidence

interval

is equal to the same mean value again

9996

0.21 minus 3.26 the same value we've

worked out up here

and that is equal to let me work that

out

996.21 minus this time

3.26 which is equal to

two 992.8 nine five so my confidence

interval then for the mean

is my lower limit is um nine nine

um two point nine five up to

my upper limit here 999.47

so i can now say with 95 confidence

that the population mean lies between

these two values

uh if i wanted to do 99 confidence the

only thing that's going to change is

this

um t value over here it will be a higher

value for a 99

confidence and this margin here will

will will

widen so our ci here is

equal to plus or minus 3.2

so that's this value here

now so that's how you calculate the

confidence interval for a sample i hope

you found this video useful

thank you for your attention