>>We’ve talked about hypothesis testing for a population mean, and we’ve gone through
some examples where we drew conclusions about hypotheses with both the critical value approach
and the p-value approach. Both approaches give the same ultimate answer regarding whether
we reject or fail to reject the null, but they come from different directions.
We’ll now apply both these approaches to hypotheses that deal with population proportions.
That is, there’s a proportion of successes that exist in the population, and we want
to answer a hypothesis about the proportion of successes. Note that, much like confidence
intervals for population proportions, we assume that the population contains two categories:
successes and failures. Remember that “successes” doesn’t necessarily imply a good thing – it’s
just a way for us to characterize two different categories such that one is the complement
of the other.
We may ask a question like: Is the proportion of defective circuits greater than point oh
five? We can collect a sample of circuits, and each of those circuits will fall into
one of two categories: defective or not defective. We can then estimate the population proportion
with the sample proportion, p-hat. And then ultimately compare p-hat to the hypothesized
proportion point oh five with a test statistic.
Generally, the notation for the hypothesized value is p_naught or p_zero. We may want to
know if the proportion is equal to p_naught or if the two are unequal. Or we could have
greater than or equal to p_naught in the null and less than p_naught in the alternative.
Or we could have less than or equal to in the null and greater than in the alternative.
Much like hypothesis tests for a population mean, the alternative hypothesis gives us
an idea of where the rejection region is: two tails, lower tail, or upper tail.
So what does that test statistic look like? Recall from our calculation of the confidence
interval that we always use the standard normal distribution for population proportions, assuming
that np and n times one minus p are both greater than four.
The test statistic measures how different the point estimate, p-hat, is from the hypothesized
value, p_naught, in terms of standard deviation units. Here we get the standard deviation
from the binomial distribution, and we use p_naught in its calculation. That is, if the
population proportion is indeed p_naught, then the standard deviation of the sampling
distribution of p-hat would be the square root of p_naught times one minus p_naught
divided by n. Therefore, the z_naught test statistic is p-hat minus p_naught divided
by the square root of p_naught times one minus p_naught over n.
The critical value approach works the same way we saw previously. We state the hypotheses,
determine the appropriate distribution (in this case, the Z distribution), we determine
where the rejection region or regions are and what the associated critical values are,
we calculate the test statistic z_naught, we compare z_naught to the critical value
or values, then we draw a conclusion relative to the null hypothesis. Same exact idea, except
now our hypotheses deal with a population proportion.
Let’s take a look at an example. At Kruger Industrial Smoothing, ball bearings undergo
four operations, the last of which is polishing, and then inspection takes place. The inspection
process determines whether or not the bearing should be repolished. A polishing machine
will be qualified for use if it can be shown to produce less than 8% of bearings that need
repolishing. In a random sample of 300 parts, 12 were defective. Based on this sample, can
the machine be qualified at 95% confidence?
The statement we focus on is “less than 8%.” Less than implies strictly less than
– no equality. We know that the equality statement must be in the null hypothesis.
Therefore, less than must be in the alternative. The null hypothesis is p greater than or equal
to point oh eight, and the alternative hypothesis is p less than point oh eight.
This is a one-tailed test with all of alpha equals point oh five in the lower tail. We
use the Z table to find the critical value. We want the tail area, but the Z table provides
the area between zero and the value z. So we need to look for point five minus point
oh five, which is point four five in the table, and look for the z value associated with this
area. The z value is between one point six four and one point six five, so let’s just
say one point six four five. Therefore, the critical value that’s in the lower tail
is negative one point six four five.
Now let’s calculate the test statistic. p-hat is 12 repolished bearings divided by
300 total bearings, or point oh four. The test statistic is point oh four minus point
oh eight divided by the square root of point oh eight times one minus point oh eight over
three-hundred. z_naught is negative two point five five.
We compare this to the critical value. z_naught of negative two point five five lies in the
rejection region. That is, it lies beyond the critical value of negative one point six
four five. As such, we reject the null and conclude that the polishing machine produces
fewer than 8% of bearings that need repolishing. We draw this conclusion with 95% confidence.
We can use this machine in the polishing process.
Let’s take a look at another example and use the p-value approach. Say you’re a usability
engineer at Initech, and you are observing the time required to perform a repetitive
task, measured in seconds. Say that you prefer that the task take 70 seconds or less. Is
the percentage of observations taking seventy or fewer seconds more than eighty percent?
We’ve got quite a few quantities going on here, so let’s first focus on the hypotheses.
Is the percentage of observations taking seventy or fewer seconds more than eighty percent?
That is, is the percentage more than eighty percent? Or, is the proportion more than point
eight. This statement refers to strictly more than. No equality. So greater than point eight
must be in the alternative. Since equality is always in the null, p less than or equal
to point eight must be the null hypothesis statement.
From our sample of sixty-four observations of the task completion time, we find that
55 observations took 70 seconds or less. Therefore, p-hat is fifty-five divided by sixty-four,
or point eight five nine. The z statistic is point eight five nine minus point eight
divided by the square root of point eight times one minus point eight divided by sixty-four,
or one point one eight.
With the p-value approach, we find the area in the tail associated with the test statistic
directly. Since this is an upper tail test, we find the area above the test statistic.
And since this is a one-tailed test, this area is entirely the p-value. If we use the
Z table, we find that the area in the tail above one point one eight is point one one
nine.
We didn’t specify a particular alpha, but we see that the p-value of point one one nine
is pretty large. It’s larger than most values of alpha we’d typically consider. Since
the p-value is larger than alpha, we fail to reject the null. We can’t conclude that
the percentage of tasks taking seventy seconds or less is greater than eighty percent. Even
though our sample percentage was eighty-five point nine. Point eight five nine and point
eight just aren’t different enough from each other, when compared to the variability
and sample size, for us to conclude that the proportion is greater than point eight.
So, we’ve applied the critical value approach and the p-value approach to examples of hypothesis
testing about population proportions. Again, the procedures don’t change from one population
parameter to the next. The test statistic changes, and maybe the distribution changes,
but the overall approach doesn’t change at all. Once you get it down the first time,
you can keep applying the idea to population means, proportions, and variances.