Developing the Informal confidence Interval .....

So far......

Given our understanding if sampling variation we know that it wouldn't make sense to use our sample median as a point estimate for the population median.

Creating an interval that we are confident contains the true population median sounds like a better idea.

One way to look at an interval....

We could take a large number of samples from the population and plot all the different sample medians on a graph (example to the left). This is showing us the amount of sampling variation present.

In this example, we could use this information to suggest the population median weight for Kiwi is likely to be between 2.46kg and 2.60kg

We cut off the few samples at the bottom and top so that the interval can be a bit narrower. This is why we said 'likely to be between' when giving our interval.

But in the real world we would never take lots of different samples, we only take ONE SAMPLE.....

The important thing from the situation above is the understanding of how wide the interval needs to be. In this case the width is 0.14kg (2.60kg - 2.46kg). This is based on how much sampling variation is present and the cutting off of the few extreme samples at the top and bottom. If we knew this width we could put it around our individual sample median (from our single sample) to create an interval that is likely to include the true population median.

So putting everything together, we have discovered that when formulating an interval from ONE SAMPLE we must take into account........

Sample Size

A larger sample size results in less sampling variation. This means that as the sample size becomes larger the interval required becomes narrower.

Spread in the POPN

The more spread in the population the more sampling variation we see. This means that as the spread becomes larger the interval required becomes larger.

Works about 90% of the time

If we wanted the interval to work 100% of the time the interval would have to be quite wide to work for the extreme sample medians. We make it narrower, and therefore more useful, by having it work about 90% of the time.

Informal Confidence Interval formula

The plus/minus bit

is added on/subtracted off the sample median to create two ends of the Confidence interval

Notice what it is made up of......

Let's test it.......

The sample size (n). On the bottom of the fraction, as it gets larger the plus/minus bit gets smaller - creating a narrower interval
The sample IQR (our best estimate for the spread in the POPN). On the top of the fraction, as it gets larger the plus/minus bit also gets larger - creating a wider interval
'1.5 x' This part is what makes the Confidence interval work about 90% of the time. This means that if we took 100 different samples from a population and created a Confidence interval with each one, we would find that about 90 would include the true population median and about 10 wouldn't.

Activity 4: How often does the Informal Confidence Interval work?

Teacher notes here

We are again going to take repeated samples from the Kiwi population. This is not something we would do in a real world situation, we are using it to understand how often the Informal Confidence Interval works.

Using the tab above, take a sample of size 150 from the Kiwi population.

Note down the sample median and the sample IQR (UQ - LQ).
Use these values, along with n = 150, to create an Informal Confidence Interval using the formula above. Round your Upper and Lower interval values to 3 dp (Creating Confidence Interval using formula video)
Open the class spreadsheet and enter your interval into one of the 100 slots on the sheet.
Repeat this process 3-4 times until all 100 slots have been filled. (Getting Confidence interval values from Inzight Lite video)

What do you notice about how many worked (green) and how many didn't (red)?

What type of sample must you have selected if you got an interval that didn't work (red)?

Make copies of the necessary documents by clicking below....

Student Document A

Using ICI formula

Student Document B

Using ICI values from InZight Lite

Class spreadsheet

Slideshow to assist with running the activity....

Slideshow

Interpreting your confidence interval

Once our informal confidence interval is calculated we must explain what it means in the context of the situation. This is called 'Interpreting your confidence interval'

To make sure the context is clear we must include the population parameter (usually MEDIAN) and the variable of interest. Remember we are suggesting where we think the population median is likely to lie so it must be made clear that you are talking about the population.

Lastly, the informal confidence interval works most of the time (around 90%) but not all of the time. We need to acknowledge this in our interpretation, this is done by beginning with the words....."I am pretty sure...."

Confidence Interval Interpretation structure

From my informal confidence interval, I am pretty sure that the [parameter] [variable] of [population] is somewhere between 2.39kg and 2.57kg.

Example.....

Confidence Interval Interpretation

From my informal confidence interval, I am pretty sure that the median weight of Kiwi in NZ is somewhere between 2.39kg and 2.57kg.

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