Welcome!
In this video I will be estimating a population proportion by constructing confidence intervals.
A point estimate (that is, a single value estimate)
for the unknown population proportion is the value of the sample proportion p-hat
which is defined by x/n
where x is the number of successes in a sample of size n.
The remaining n – x are considered failures.
The sample proportion of failures, 1 – p-hat, is often denoted as q-hat.
Suppose a random sample of 500 adults responded to a yes or no question.
If 420 responded yes, then the sample proportion is 420/500
which is 0.84.
This 0.84 is a point estimate for the population proportion.
It gives us an idea of what the population proportion might be.
However, it does not tell us that 84% of the population will respond “yes”.
To infer or generalize about the population
we construct a confidence interval that is, a range of values that
we expect to capture the population parameter, p at some confidence level.
The confidence level represents the proportion of times we expect to capture
the population parameter with repeated sampling
The confidence interval for p is computed using this formula.
The sample proportion plus or minus the margin of error
The margin of error is the z-critical value times the standard error of the sample proportion.
The standard error is the estimated standard deviation
of the distribution of sample proportions. That is, an estimation of how much on average
we expect a sample proportion to differ from the true population proportion
This confidence interval is valid when the sample size is large enough
for the minimum required number of successes and failures
which could be 5 or more depending on your discipline
The z-critical value is determined based on the required confidence level.
z-critical values for commonly used confidence intervals are shown here.
However, most confidence intervals you see in reports are constructed
at the 95% confidence level unless otherwise stated.
But let’s construct intervals at these unconventional confidence levels.
given n = 500 and p̂ = 0.84 Here again is the confidence interval formula.
Let’s first calculate the standard error Substituting the values gives 0.0164
This value is typically very small so it is important not to round intermediate calculations.
Now, for the 85% confidence interval the area in the tails is 1 – 0.85
which equals 0.15. This 0.15 is called the significance level
represented by alpha That is, alpha equals 1 minus the confidence level
The critical value is the z-value separating the confidence region from the tail area.
The tail area is called the critical region. The area in each tail is alpha/2
which is 0.15/2 or 0.075. From the less-than z-table
the closest area in the table to 0.075 is 0.0749 which corresponds to a z-score of -1.44
So, separating the left tail here we have z critical value of -1.44
and +1.44 on the right here due to symmetry So we say we have a critical value z of 1.44
Now, plugging that into the formula, we obtain a margin of error of .0236,
a lower limit of 0.816 and an upper limit of 0.864.
To interpret that we say we are 85% confident
that the true population lies between 0.816 and .864
What the confidence interval really means is that if we construct 85% confidence intervals (like
this one) for many random samples from this population
in the long run, we expect about 85% of those intervals
to contain the true population proportion and about 15% not to contain it.
This interval is just one of those many intervals. It could belong in the 85% or in the 15%.
In essence, this interval may or may not contain the true population proportion.
For a 94% confidence interval alpha is 1 – 0.94 which is 0.06
and alpha/2 is 0.03 The closest area to 0.03 is 0.0301
which corresponds to a z-score of -1.88
So, the 94% confidence z-critical value has an absolute value of 1.88
Plugging that into the formula we have a margin of error of 0.0308
a lower limit of 0.809 and an upper limit of 0.871.
And for the 97% confidence level, alpha is 0.03 the area in each tail is 0.015
which we find exactly in the table And that corresponds to a
critical z-score of 2.17. Plugging that into the formula
produces a margin of error of 0.0356, a lower limit of 0.804, and an upper
limit of 0.876. That is, we are
97% confident that the unknown population proportion lies between 80.4% and 87.6%.
From the results we see that as confidence level increases
the z-critical value increases which increases the margin of error
and in turn makes the confidence interval wider. Conversely, decreasing the confidence level
decreases the width of the confidence interval. Apart from the confidence level
the sample size also affects the width of the confidence interval.
Since the sample size is in the denominator of the standard error formula,
increasing the sample size will decrease the size of the standard error, which will in turn
decrease the margin of error, and consequently reduce the width of the confidence interval.
This agrees with the logic that larger samples are expected to produce more precise estimation.
So a sample of size 200 will produce a wider confidence interval than a sample of size 500
but narrower than a sample size of 100. And that’s it for this video.
Thanks for watching.