Solved – Choice of null and alternative hypothesis

Question

A firm producing tobacco cigarettes claims that it has discovered a new technique for curing tobacco leaves, that results in an average nicotine content of a cigarette of less than 1.5 mg. To test this claim, a sample of 20 of the firm's cigarettes are analyzed. If it were known that the standard deviation of a cigarette's nicotine content was 0.7 mg, what conclusions can be drawn, at the 5% significance level, if the average nicotine content of these 20 cigarettes were 1.42 mg?

According to the text, the null and alternative hypothesis are:

$H_0$:μ≥1.5
$H_a$:μ<1.5

The z statistic is -0.511 and p{Z<=-0.511) = 0.305. Since this exceeds 0.05, $H_0$ is not rejected. That is, we don't reject the claim that the average is >= 1.5.

What if the firm had claimed that their research resulted in an average of less than or equal to 1.5mg? I've read that the null hypothesis has to contain the equality. So the new hypotheses will be:

$H_0$:μ<=1.5
$H_a$:μ>1.5

The z statistic remains the same P{Z>=0.511) = .695. Thus $H_0$ is again not rejected. So this time, we are not rejecting the claim that the average is less than or equal to 1.5, which is the result that the firm must be hoping for. So what is going on here? Can including a single value in the claim make this bizarre difference? Can't they just pretend that they never carried out the first test, and then just do the second, getting the proof that they're looking for?

Answer 1

As explained in this answer What follows if we fail to reject the null hypothesis? one can only find 'statistical evidence' for $H_a$ if the sample is such that $H_0$ is rejected. If $H_0$ can not be rejected then the only conclusion one can draw is that $H_0$ does not lead to a statistical contradiction.

In other words if one can not reject $H_0$ that does not imply that $H_0$ is true, this is the reason why we say that it is ''accepted'' (which is not the same as statistically proven).

In your example, the first hypothesis test $H_0: \mu \le 1.5$ versus $H_a: \mu > 1.5$ you find a p-value of 0.305 which means that your sample does not provide ''evidence'' for $H_a$. This does however not imply that you can conclude that $H_0$ is true (it is only not rejected or accepted, but there is no evidence for $H_0$.

In your second situation you test $H_0: \mu \le 1.5$ versus $H_a: \mu > 1.5$ you find a p-value of 0.695, so you have no evidence thet $H_a$ is true. The fact that you can not prove that $H_a$ holds does not imply that $H_0$ holds.

The only conclusion that you can draw from your example is that your sample does not provide evidence for $H_a: \mu < 1.5$ nor does it provide evidence for $H_a: \mu > 1.5$.