Up until now, we have used the critical region approach in conducting our hypothesis tests. Now, let's take a look at an example in which we use what is called the ** P-value approach**.

Among patients with lung cancer, usually 90% or more die within three years. As a result of new forms of treatment, it is felt that this rate has been reduced. In a recent study of *n* = 150 lung cancer patients, *y* = 128 died within three years. Is there sufficient evidence at the *α* = 0.05 level, say, to conclude that the death rate due to lung cancer has been reduced?

**Solution.** The sample proportion is:

\[\hat{p}=\dfrac{128}{150}=0.853\]

The null and alternative hypotheses are:

*H*_{0}: *p* = 0.90 and *H _{A}*:

The test statistic is, therefore:

\[Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{0.853-0.90}{\sqrt{\dfrac{0.90(0.10)}{150}}}=-1.92\]

And, the rejection region is:

Since the test statistic *Z* = −1.92 < −1.645, we reject the null hypothesis. There is sufficient evidence at the *α* = 0.05 level to conclude that the rate has been reduced.

What if we set the significance level *α* = *P*(Type I Error) to 0.01? Is there still sufficient evidence to conclude that the death rate due to lung cancer has been reduced?

**Solution.** In this case, with *α *= 0.01,* *the rejection region is *Z* ≤ −2.33. That is, we reject if the test statistic falls in the rejection region defined by *Z* ≤ −2.33:

Because the test statistic *Z* = −1.92 > −2.33, we do not reject the null hypothesis. There is insufficient evidence at the *α* = 0.01 level to conclude that the rate has been reduced.

In the first part of this example, we rejected the null hypothesis when *α* = 0.05. And, in the second part of this example, we failed to reject the null hypothesis when *α* = 0.01. There must be some level of *α, *then, in which we cross the threshold from* *rejecting to not rejecting the null hypothesis. What is the smallest *α*−level that would still cause us to reject the null hypothesis?

**Solution.** We would, of course, reject any time the critical value was smaller than our test statistic −1.92:

That is, we would reject if the critical value were −1.645, −1.83, and −1.92. But, we wouldn't reject if the critical value were −1.93. The *α*−level associated with the test statistic −1.92 is called the ** P-value**. It is the smallest

*P*(*Z* < −1.92) = 0.0274

So far, all of the examples we've considered have involved a one-tailed hypothesis test in which the alternative hypothesis involved either a less than (<) or a greater than (>) sign. What happens if we weren't sure of the direction in which the proportion could deviate from the hypothesized null value? That is, what if the alternative hypothesis involved a not-equal sign (≠)? Let's take a look at an example.

What if we wanted to perform a "**two-tailed**" test? That is, what if we wanted to test:

*H*_{0}: *p* = 0.90 versus *H*_{A}: *p* ≠ 0.90

at the *α* = 0.05 level?

**Solution.** Let's first consider the **critical value approach**. If we allow for the possibility that the sample proportion could either prove to be too large or too small, then we need to specify a threshold value, that is, a critical value, in each tail of the distribution. In this case, we divide the "**significance level**" *α* by 2 to get *α*/2:

That is, our rejection rule is that we should reject the null hypothesis *H*_{0} if *Z* ≥ 1.96 or we should reject the null hypothesis *H*_{0 }if *Z* ≤ −1.96. Alternatively, we can write that we should reject the null hypothesis *H*_{0} if |*Z*| ≥ 1.96. Because our test statistic is −1.92, we just barely fail to reject the null hypothesis, because 1.92 < 1.96. In this case, we would say that there is insufficient evidence at the *α* = 0.05 level to conclude that the sample proportion differs significantly from 0.90.

Now for the ** P-value approach**. Again, needing to allow for the possibility that the sample proportion is either too large or too small, we multiply the

That is, the *P*-value is:

\[P=P(|Z|\geq 1.92)=P(Z>1.92 \text{ or } Z<-1.92)=2 \times 0.0274=0.055\]

Because the *P*-value 0.055 is (just barely) greater than the significance level *α* = 0.05, we barely fail to reject the null hypothesis. Again, we would say that there is insufficient evidence at the *α* = 0.05 level to conclude that the sample proportion differs significantly from 0.90.

Let's close this example by formalizing the definition of a *P*-value, as well as summarizing the *P*-value approach to conducting a hypothesis test.

Alternatively (and the way I prefer to think of |

If the *P*-value is small, that is, if *P* ≤ *α*, then we reject the null hypothesis *H*_{0}.

By the way, to test *H*_{0}: *p* = *p*_{0}, some statisticians will use the test statistic:

\[Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}}\]

rather than the one we've been using:

\[Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}\]

One advantage of doing so is that the interpretation of the confidence interval — does it contain *p*_{0}? — is always consistent with the hypothesis test decision, as illustrated here:

For the sake of ease, let:

\[se(\hat{p})=\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\]

**Two-tailed test.** In this case, the critical region approach tells us to reject the null hypothesis *H*_{0}:* p* = *p*_{0} against the alternative hypothesis *H*_{A}: *p* ≠ *p*_{0}:

if \[Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \geq z_{\alpha/2}\] or if \[Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \leq -z_{\alpha/2}\]

which is equivalent to rejecting the null hypothesis:

if \[\hat{p}-p_0 \geq z_{\alpha/2}se(\hat{p})\] or if \[\hat{p}-p_0 \leq -z_{\alpha/2}se(\hat{p})\]

which is equivalent to rejecting the null hypothesis:

if \[p_0 \geq \hat{p}+z_{\alpha/2}se(\hat{p})\] or if \[p_0 \leq \hat{p}-z_{\alpha/2}se(\hat{p})\]

That's the same as saying that we should reject the null hypothesis *H*_{0} if *p*_{0} is not in the (1−*α*)100% confidence interval!

**Left-tailed test.** In this case, the critical region approach tells us to reject the null hypothesis *H*_{0}:* p* = *p*_{0} against the alternative hypothesis *H*_{A}: *p* < *p*_{0}:

if \[Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \leq -z_{\alpha}\]

which is equivalent to rejecting the null hypothesis:

if \[\hat{p}-p_0 \leq -z_{\alpha}se(\hat{p})\]

which is equivalent to rejecting the null hypothesis:

if \[p_0 \geq \hat{p}+z_{\alpha}se(\hat{p})\]

That's the same as saying that we should reject the null hypothesis *H*_{0} if *p*_{0} is not in the upper (1−*α*)100% confidence interval:

\[(0,\hat{p}+z_{\alpha}se(\hat{p}))\]