The original

plan w

as to

use the same

triglyceride

data we used for the previous assignment. We

would

treat the 3026 women in the data set as our population. We

would then

take a sample from this

population and use it to estimate the mean triglyceride level for the population. We

would

run a

simulation with repeated samples and

will

see

whether 95% of 95% confidence intervals actually do

capture

the population mean

(

= 116.9)

.

Finally we

would

examine the eff

ect of changing the

confidence level.

The pr

oblem is that not all s

tudents were able to download Fathom onto their lap

tops

. So instead of

using Fathom to perform a simulation, we

ll use the appl

et on the textbook website

to explore

confidence

levels.

Go to the

Confid

ence Interval

applet

. There is also a link to the appl

et on our Canvas pa

ge.

In this applet we construct confidence intervals for the mean () of a

n

orm

al population distribution.

Each interval is based on a SRS of size

n

. The dot marks the sample mean, which is the center of the

interval. The lines on each side of the dot span the confidence interval. The total number of SRSs, the

number that hit (i.e.

, the confidence interval contained ), and the percent hit are tallied for you

.

The de

tails of th

e simulation will be handle

d in

the

backgrou

nd, but w

e can still get an idea of how

sampling variability of the s

ample mean effects

Generating a single confidence interval

1.

Usin

g the default settings in the

Confidence Interval

applet (95% confidence level and

n

= 20),

click

SAMPLE

to choose an SRS and display its confidence interval.

O

n the right

you

ll see the

s

ampled values as

small yellow dots; the lar

ge do

t will

show the s

ample mean, and the lines on

each side of th

is dot span the

confidence inter

val.

Q1

Is the spread in the data, shown as yellow dots below the confidence interval, larger than the

span of

the confidence interval?

(This is typically

the case be

cause

the

width of the confidence interval is based on the

variability

of the s

am

ple mean. A

nd

sample means are less variable than indi

vidua

l

observations.)

2.

For the same data set, you can compare the span of the confidence interval for different values

of

C

by sliding the confidence level to a

new value.

Q2

For the SRS you generated in

S

tep 1

,

what happens to the span of the interval when you

move

C

to 99%? What about 90%? Describe the relationship you find between the confidence

level

C

and the span of the confidence interval.

Capture rate of multiple confidence

intervals

The idea of an 80% confidence interval is that the inter

val captures the true parameter value in 80% of

all samples. Thats not high enough confidence for practical use, but 80% hits and 20% misses make it

easy to see how a confidence interval behaves in repeated samples from the same population.

3.

Set the confid

ence level in the

Confidence Interval

applet to 80%.

Click

RES

E

T

and the

n

c

lick

SAMPLE 25

to take 25 samples all at one. Intervals that

contain

the population

mean (

hits

)

will be colored gray;

misses

will be colored red. Click on any confidence interval to show the

sample

data th

at the i

nterval i

s

based on.

Note that any intervals

that

missed

had a sample mean that was

un

usually high or low.

Whether a co

nfidence interval contains

the

p

opulation mean depends sole

l

y

on

the

value of the

sample mean.

Q3

How many of the 25 intervals contain the true mean

? What proportion contain the true mean?

Include a screenshot of th

e resul

ts so I can see that you attempted th

is.

4.

Click

SAMPLE 25

again to get the confidence intervals from 50 SRSs.

Keep clicking

SAMPLE 25

until you have 100 samples

(Total: 100)

and r

ecord the proportion of hits among 100

SRSs.

Q

4

How many of the

100

intervals do you

expect

to include the population mean How many of the

intervals

actually

include the

p

opulation mean?

A

s

the number of samples increases, we expect the percent of captures to get closer to the confidence

level, 80%. Do you find this pattern in your results?

5.

Change

the

confidence le

vel to 95%.

Click

RESET

then c

lick

SAMPLE 25

to take 25 samples all at

on

c

e.

Click

SAMPLE 25

3 more times so

you have 100 samples.

(Total: 100)

Include a screenshot

of th

e resul

ts so I can see that you attempted th

is.

Q5

How many of the

100

intervals do you

expect

to include the population mean How many of the

intervals

actually

include the

p

opulation mean?

For the simulation, the appl

et kn

ew w

hat

was, so it could determine which intervals hit and wh

ich

ones missed.

I

n practice w

e cant determine whether

one

new S

RS will result in an interval that

contains

or not

, sinc

e

is generally u

nknown

.

The confidence level only tells us what percent will

contain

in the long run.

(

Note:

It is understandable if the

two numbers

in Q4

and

in

Q

5

a

re not exactly

equal

,

as 100 i

s

not

really

large enough to

be considered

the long run

as far as simulations

go.

However

,

t

he numbers should be

fa

irly close.)