This lesson will show you
how to use desmos.com
to construct a confidence interval of a proportion.
A smartphone manufacturer is interested
in constructing a 95% confidence interval
for the proportion of smartphones
that break before the warranty expires.
91 of the 1,610 randomly selected smartphones
broke before the warranty expired.
Round answers to four decimal places where possible.
Let's first determine the sample proportion,
which is X divided by N or in our case,
91 divided by 1,610.
As a decimal, this is approximately 0.0565.
The sample size N is 1,610.
And now to determine the Z score,
or Z sub alpha divided by two,
that corresponds to the Z score
we need for the 95% confidence interval.
Alpha divided by two
is the area to the right of the Z score,
which you can see from the graph is 0.025 as a decimal.
To find the Z score using Desmos though,
we need the area to the left of the Z score, which is 0.975.
Let's go to Desmos and determine the Z score.
To do this, click show keypad, click Functions.
Under the distributions menu,
click normaldist for normal distribution.
For the standard normal distribution,
we enter a mean of zero, a standard deviation of one,
close parenthesis, period or decimal point,
click functions, click inversecdf.
Then enter the area to the left of the Z score,
which again is 0.975.
We can see the Z score is approximately 1.96.
Now that we have all the information we need,
we'll enter the error formula into Desmos,
then enter the Z score,
the sample proportion and the sample size.
Once we determine E,
we can determine the confidence interval
by adding and subtracting E from the sample proportion,
which is 0.0565.
So going back to Desmos, in cell two,
we enter E equals Z square root, open parenthesis P,
open parenthesis one minus P, close parenthesis,
close parenthesis divided by N.
Notice below, we have an option
to add sliders for Z, P and N.
Click all.
And now we enter Z, P and N to determine the error.
Let's close the Desmos keypad.
The Z score is 1.96.
The sample proportion is 0.0565,
and N is 1,610.
Notice how the error to four decimal places is 0.0113.
So again, this is the value we subtract
and add to the sample proportion P.
Let's do this in cell six and seven.
In cell six, we have 0.0565 minus 0.0113.
The lower bound of the 95% confidence interval is at 0.0452.
In cell seven, we enter 0.0565 plus 0.0113.
The upper bound of the 95% competence interval is 0.0678.
Let's go ahead and record this.
For part A, with 95% confidence
the proportion of all smartphones
that break before the warranty expires
is between 0.0452 and 0.0678.
[indistinct] percentages would be 4.52% and 6.78%.
And now to better understand the 95% confidence interval
for part B.
If many groups of 1,610 randomly selected smartphones
are selected, then a different confidence interval
would be produced for each group.
About 95% of these confidence intervals
will contain the true population proportion
of all smartphones that break before the warranty expires.
And about 100% minus 95% or 5%
will not contain the true population proportion.
I hope you found this helpful.