Presentation Notes:
The idea behind establishing a confidence interval is to take a sample and use its mean or proportion as a
center for the interval. We will deal with a sample proportion in this lesson. An example of this is the
president’s approval rating. It is given as a percentage (which is a proportion) and then a confidence
interval is given around it. For example, “The president’s approval rating is 46% with a margin of error
of 3%.” This actually means that .46 of the sample indicated approval of the president and that we
believe that, with a degree of certainty to be discussed later, the proportion for the whole population is
between 0.46-0.03= 0.43 and 0.46+ 0.03= 0.49.
We can model the kind of variation that would occur around an expected proportion in some situations.
For the president example, we have no advanced expectation. If we were flipping a coin, however, the
expected proportion of heads would be 0.5. We can use this as a baseline for our simulation, which is
referred to as the null hypothesis in statistics. Since not all samples of a certain number of flips would be
exactly 0.5 heads, we can simulate a sample of a given number of flips using technology or manipulatives
that appropriately model the baseline proportion. We can do this repeatedly, creating multiple samples
and recording the proportion of each sample. Then, we can generate a dot plot of the sample proportions.
With a large number of samples, this dot plot will begin to approximate a normal curve. This normal
curve, created by many samples, is called the sampling distribution. That is, it is the distribution of
proportions for many samples. This is called a “bootstrap” sampling distribution. (In actuality, if we were
conducting a survey such as the presidential approval rating, the sampling distribution would be the
proportion of not “many” samples but every possible sample from the entire population.) Once we have a
bootstrap sampling distribution, we can use it to make inferences about the population.
We can first decide if the sample represents a significant departure from the null hypothesis. This is often
tested at the 5% level (the level chosen is referred to as the  level). That is, we find the proportions that
are outside the interval where 95% or 0.95 of the samples in the bootstrap normal distribution reside.
This can be done manually by counting the number of samples and locating the symmetric region around
the null hypothesis proportion where 0.95 of samples lie. If our sample proportion is in this interval, this
gives us no evidence to refute the null hypothesis. If our sample is outside this interval, then we can
conclude, with 95% certainty, that the null hypothesis is probably not true. Samples this far away from
the null hypothesis would only occur 5% of the time if the null hypothesis were accurate.
We can also use the interval about the null hypothesis to establish a rough approximate of a confidence
interval (recall the margin of error in the president’s approval rating example of 0.03 ). With some loss
of precision but still with reasonable accuracy, we can slide the interval to be centered about the sample
mean to establish an interval in which we are 95% certain contains the proportion for the population.