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Proportions (Step 3)

CO-4: Distinguish among different measurement scales, choose the appropriate descriptive and inferential statistical methods based on these distinctions, and interpret the results.
LO 4.33: In a given context, distinguish between situations involving a population proportion and a population mean and specify the correct null and alternative hypothesis for the scenario.
LO 4.34: Carry out a complete hypothesis test for a population proportion by hand.
CO-6: Apply basic concepts of probability, random variation, and commonly used statistical probability distributions.
LO 6.26: Outline the logic and process of hypothesis testing.
LO 6.27: Explain what the p-value is and how it is used to draw conclusions.
Video: Proportions (Step 3) (14:46)

Step 3. Finding the P-value of the Test

So far we’ve talked about the p-value at the intuitive level: understanding what it is (or what it measures) and how we use it to draw conclusions about the statistical significance of our results. We will now go more deeply into how the p-value is calculated.

It should be mentioned that eventually we will rely on technology to calculate the p-value for us (as well as the test statistic), but in order to make intelligent use of the output, it is important to first understand the details, and only then let the computer do the calculations for us. Again, our goal is to use this simple example to give you the tools you need to understand the process entirely. Let’s start.

Recall that so far we have said that the p-value is the probability of obtaining data like those observed assuming that Ho is true. Like the test statistic, the p-value is, therefore, a measure of the evidence against Ho. In the case of the test statistic, the larger it is in magnitude (positive or negative), the further p-hat is from p0, the more evidence we have against Ho. In the case of the p-value, it is the opposite; the smaller it is, the more unlikely it is to get data like those observed when Ho is true, the more evidence it is against Ho. One can actually draw conclusions in hypothesis testing just using the test statistic, and as we’ll see the p-value is, in a sense, just another way of looking at the test statistic. The reason that we actually take the extra step in this course and derive the p-value from the test statistic is that even though in this case (the test about the population proportion) and some other tests, the value of the test statistic has a very clear and intuitive interpretation, there are some tests where its value is not as easy to interpret. On the other hand, the p-value keeps its intuitive appeal across all statistical tests.

How is the p-value calculated?

Intuitively, the p-value is the probability of observing data like those observed assuming that Ho is true. Let’s be a bit more formal:

  • Since this is a probability question about the data, it makes sense that the calculation will involve the data summary, the test statistic.
  • What do we mean by “like” those observed? By “like” we mean “as extreme or even more extreme.”

Putting it all together, we get that in general:

The p-value is the probability of observing a test statistic as extreme as that observed (or even more extreme) assuming that the null hypothesis is true.

By “extreme” we mean extreme in the direction(s) of the alternative hypothesis.

Specifically, for the z-test for the population proportion:

  1. If the alternative hypothesis is Ha: p < p0 (less than), then “extreme” means small or less than, and the p-value is: The probability of observing a test statistic as small as that observed or smaller if the null hypothesis is true.
  2. If the alternative hypothesis is Ha: p > p0 (greater than), then “extreme” means large or greater than, and the p-value is: The probability of observing a test statistic as large as that observed or larger if the null hypothesis is true.
  3. If the alternative is Ha: p ≠ p0 (different from), then “extreme” means extreme in either direction either small or large (i.e., large in magnitude) or just different from, and the p-value therefore is: The probability of observing a test statistic as large in magnitude as that observed or larger if the null hypothesis is true.(Examples: If z = -2.5: p-value = probability of observing a test statistic as small as -2.5 or smaller or as large as 2.5 or larger. If z = 1.5: p-value = probability of observing a test statistic as large as 1.5 or larger, or as small as -1.5 or smaller.)

OK, hopefully that makes (some) sense. But how do we actually calculate it?

Recall the important comment from our discussion about our test statistic,

ztestprop

which said that when the null hypothesis is true (i.e., when p = p0), the possible values of our test statistic follow a standard normal (N(0,1), denoted by Z) distribution. Therefore, the p-value calculations (which assume that Ho is true) are simply standard normal distribution calculations for the 3 possible alternative hypotheses.

Alternative Hypothesis is “Less Than”

The probability of observing a test statistic as small as that observed or smaller, assuming that the values of the test statistic follow a standard normal distribution. We will now represent this probability in symbols and also using the normal distribution.

image258

A normal distribution curve (N(0,1)). Marked on the horizontal axis are z-scores of 0 and z. z is to the left of 0 because it is for a test statistic which is smaller than p_0. The p-value is the area to the left of z under the curve.

Looking at the shaded region, you can see why this is often referred to as a left-tailed test. We shaded to the left of the test statistic, since less than is to the left.

Alternative Hypothesis is “Greater Than”

The probability of observing a test statistic as large as that observed or larger, assuming that the values of the test statistic follow a standard normal distribution. Again, we will represent this probability in symbols and using the normal distribution

image260A normal distribution curve (N(0,1)). Marked on the horizontal axis are z-scores of 0 and z. z is to the right of 0 because it is for a test statistic which is larger than p_0. The p-value is the area to the right of z under the curve.

Looking at the shaded region, you can see why this is often referred to as a right-tailed test. We shaded to the right of the test statistic, since greater than is to the right.

Alternative Hypothesis is “Not Equal To”

The probability of observing a test statistic which is as large in magnitude as that observed or larger, assuming that the values of the test statistic follow a standard normal distribution.

image262A normal distribution curve (N(0,1)). Marked on the horizontal axis are z-scores of 0, -|z|, and |z|, where |z| and -|z| is the z-score of the observed test statistic. The p-value is the sum of the area to the right of |z| under the curve and the area to the left of -|z| under the curve.

This is often referred to as a two-tailed test, since we shaded in both directions.

Next, we will apply this to our three examples. But first, work through the following activities, which should help your understanding.

Learn by Doing: Proportions (Step 3)
Did I Get This?: Proportions (Step 3)

EXAMPLE:

Has the proportion of defective products been reduced as a result of the repair?

A large circle represents the population of products produced by the machine (following the repair). We want to know p about this population, or what is the proportion of defective products. The two hypotheses are H_0: p = .20 and H_a: p < .20. We take a sample of 400 products, represented by a smaller circle. We find that 64 of these are defective. p-hat = 64/400 = .16, and z = -2.

The p-value in this case is:

  • The probability of observing a test statistic as small as -2 or smaller, assuming that Ho is true.

OR (recalling what the test statistic actually means in this case),

  • The probability of observing a sample proportion that is 2 standard deviations or more below the null value (p0 = 0.20), assuming that p0 is the true population proportion.

OR, more specifically,

  • The probability of observing a sample proportion of 0.16 or lower in a random sample of size 400, when the true population proportion is p0 =0.20

In either case, the p-value is found as shown in the following figure:

A normal distribution curve (N(0,1)). Marked on the horizontal axis are z-scores of 0 and z. z is to the left of 0 because it is for a test statistic which is smaller than p_0. The p-value is the area to the left of z under the curve.

To find P(Z ≤ -2) we can either use the calculator or table we learned to use in the probability unit for normal random variables. Eventually, after we understand the details, we will use software to run the test for us and the output will give us all the information we need. The p-value that the statistical software provides for this specific example is 0.023. The p-value tells us that it is pretty unlikely (probability of 0.023) to get data like those observed (test statistic of -2 or less) assuming that Ho is true.

EXAMPLE:

Is the proportion of marijuana users in the college higher than the national figure?

A large circle represents the population Students at the college. We want to know p about this population, or what is the population proportion of students using marijuana. The question we wish to answer is "is p .157 (like the national figure) or higher?" We take a sample of 100 students, represented by a smaller circle. We find that 19 use marijuana. p-hat = 19/100 = .19, and z = .91

The p-value in this case is:

  • The probability of observing a test statistic as large as 0.91 or larger, assuming that Ho is true.

OR (recalling what the test statistic actually means in this case),

  • The probability of observing a sample proportion that is 0.91 standard deviations or more above the null value (p0 = 0.157), assuming that p0 is the true population proportion.

OR, more specifically,

  • The probability of observing a sample proportion of 0.19 or higher in a random sample of size 100, when the true population proportion is p0=0.157

In either case, the p-value is found as shown in the following figure:

A normal distribution curve (N(0,1)). Marked on the horizontal axis are z-scores of 0 and z. z is to the right of 0 because it is for a test statistic which is larger than p_0. The p-value is the area to the right of z under the curve.

Again, at this point we can either use the calculator or table to find that the p-value is 0.182, this is P(Z ≥ 0.91).

The p-value tells us that it is not very surprising (probability of 0.182) to get data like those observed (which yield a test statistic of 0.91 or higher) assuming that the null hypothesis is true.

EXAMPLE:

Did the proportion of U.S. adults who support the death penalty change between 2003 and a later poll?

A large circle represents the population US Adults. We want to know p about this population, which is population proportion which support the death penalty. The question we wish to answer is "has p changed since 2003 (when it was .64)?" We take a sample of 1000 US Adults, represented by a smaller circle. We find that 675 are in favor. p-hat = 675/1000 = .675, and z = 2.31

The p-value in this case is:

  • The probability of observing a test statistic as large as 2.31 (or larger) or as small as -2.31 (or smaller), assuming that Ho is true.

OR (recalling what the test statistic actually means in this case),

  • The probability of observing a sample proportion that is 2.31 standard deviations or more away from the null value (p0 = 0.64), assuming that p0is the true population proportion.

OR, more specifically,

  • The probability of observing a sample proportion as different as 0.675 is from 0.64, or even more different (i.e. as high as 0.675 or higher or as low as 0.605 or lower) in a random sample of size 1,000, when the true population proportion is p0= 0.64

In either case, the p-value is found as shown in the following figure:

A normal distribution curve (N(0,1)). Marked on the horizontal axis are z-scores of 0, -|z|, and |z|, where |z| and -|z| is the z-score of the observed test statistic. The p-value is the sum of the area to the right of |z| under the curve and the area to the left of -|z| under the curve.

Again, at this point we can either use the calculator or table to find that the p-value is 0.021, this is P(Z ≤ -2.31) + P(Z ≥ 2.31) = 2*P(Z ≥ |2.31|)

The p-value tells us that it is pretty unlikely (probability of 0.021) to get data like those observed (test statistic as high as 2.31 or higher or as low as -2.31 or lower) assuming that Ho is true.

Comment:

  • We’ve just seen that finding p-values involves probability calculations about the value of the test statistic assuming that Ho is true. In this case, when Ho is true, the values of the test statistic follow a standard normal distribution (i.e., the sampling distribution of the test statistic when the null hypothesis is true is N(0,1)). Therefore, p-values correspond to areas (probabilities) under the standard normal curve.

Similarly, in any test, p-values are found using the sampling distribution of the test statistic when the null hypothesis is true (also known as the “null distribution” of the test statistic). In this case, it was relatively easy to argue that the null distribution of our test statistic is N(0,1). As we’ll see, in other tests, other distributions come up (like the t-distribution and the F-distribution), which we will just mention briefly, and rely heavily on the output of our statistical package for obtaining the p-values.

We’ve just completed our discussion about the p-value, and how it is calculated both in general and more specifically for the z-test for the population proportion. Let’s go back to the four-step process of hypothesis testing and see what we’ve covered and what still needs to be discussed.

The Four Steps in Hypothesis Testing

  • STEP 1: State the appropriate null and alternative hypotheses, Ho and Ha.
  • STEP 2: Obtain a random sample, collect relevant data, and check whether the data meet the conditions under which the test can be used. If the conditions are met, summarize the data using a test statistic.
  • STEP 3: Find the p-value of the test.
  • STEP 4: Based on the p-value, decide whether or not the results are statistically significant and draw your conclusions in context. 
  • Note: In practice, we should always consider the practical significance of the results as well as the statistical significance.

With respect to the z-test the population proportion:

Step 1: Completed

Step 2: Completed

Step 3: Completed

Step 4. This is what we will work on next.

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