The null hypothesis should be stated in a way that it implies a specific probability distribution (called the 'null' distribution), which is used to compute the P-value.

In general, the alternative might be one-sided or two-sided.

Two-sided: against

Right-sided: (or against alternative

Left-sided: (or against alternative

Based on the information provided, you need the left-sided alternative. Notice that the null hypothesis contains an -sign in all of the formulations---whether as of [Different authors have various preferences about this.]

In any case, you should choose Answer (2).

The effect size (10% or whatever) can be determined after you have data. If you want to be reasonably sure to detect an effect of a particular size, then you need to do a 'power and sample size' procedure to see how many subjects should be used in the study.

I agree with @Pitouille that specifying the desired effect as "10%" better may lead to confusion. Ten percent of what? (More increase in red blood cells? decrease in blood sugar level? decrease in patient complaints of pain? increase in patient 'recovery"?

What will you do if the data from the study show a clear indication of "8%" better? Abandon Drug A as useless? do a new study a double the dosage? What will you do if Drug A shows slightly better results, but not by quite enough to be statistically significant at the desired level? These are important considerations that should influence the design of the study, but usually not for inclusion in the statement of

The purpose of the null is to convert a problem from one of inductive reasoning to one of deductive reasoning. The alternative, and the method that preceded it was the method of inverse probability. That method is now generally called Bayesian statistics.

Imagine that you had three scientific hypotheses, denoted a, b, and c. Imagine that the true model is d, but no one has yet to discover this. The world is still flat, white is still a color, and Mercury follows Newton's laws.

A Bayesian test would create three hypotheses, , , and . For a data set that is large enough, you would end up with the hypothesis or the combination of hypotheses that are most likely true. However, since wasn't tested, you may continue to be fooled by the idea yet to be thought of.

The Frequentist hypothesis testing regime would assume that the alternative hypothesis is , and the null is . The null contains every hypothesis that is not the alternative.

The first example in the academic literature, but not the first null hypothesis, is where R.A. Fisher assumed that Mendel's laws were false as the null. If you discredit the null, then you exclude every explanation, including those not yet considered. The first null hypothesis was that Muriel Bristol (Fisher's boss) could not correctly detect the difference between tea poured into mild from milk poured into tea. That was the very first statistical test.

There is a slight difference between R.A. Fisher's idea of a null and Pearson and Neyman's idea of a null. Fisher felt there was a null, but no alternative hypothesis. If you rejected the null, it told you what was wrong, but was not directive as to what was correct automatically. Pearson and Neyman championed acceptance and rejection regions based on frequencies, and they felt the method directed behavior.

The logic was that the method created a probabilistic version of modus tollens. Modus tollens is "if A then B; and, not B; therefore, not A." Or, if the null is true, then the test will appear in the acceptance region if it does not, then you can reject the null.

The weakness of the methodology was proposed by an author that I cannot currently locate in this somewhat tongue-in-cheek way. There are 535 elected members representing the states in the U.S. Congress. There are 360 million Americans. Therefore since 535/360000000 is less than .05, if you randomly sample a group of Americans and pick a member of Congress, they cannot be an American (p<.05).

While Fisher's no effect hypothesis is the most common version, because of its implication would be that something has an effect in the alternative, it is not a requirement that a parameter equals zero, or a set of parameters all equal zero.

What matters is that the null is the opposite of what you are wanting to assert before seeing the data.

That makes the null hypothesis method a potent tool. Think about this as a rhetorical device. Your opponent opposes that you recently believe you have discovered.

You do not assert is true. You assert your opponent's position of is true and build your probabilities on the assumption that you are the only person that is wrong. Everyone is right, and you are wrong.

It is a powerful rhetorical tool to concede the argument from the beginning, but then ask, "what would the world look like if I am the one that is wrong?" That is the null. If you reject the null, then what you are really saying is that "nature rejects all other ideas except mine."

Now as to your question, you want to show that college algebra matters, therefore your null hypothesis is that college algebra does not matter. We will ignore all the other methodological issues that would really be present since people without college algebra may have other self-selection issues as will the people with college algebra.

Your null is that algebra does not matter. The alternative is that it does. If the p-value is less than your cutoff, chosen before collecting the data, then you can reject the null. If it is not, then you should behave as if it is true until you either do more research or find another way to come to a conclusion.

It would be dubious, ignoring the methodology issues, to assert that college algebra matters as you only have one sample. The method is intended for repetition. Nonetheless, you would only be made a fool of no more than percent of the time, ignoring the methodological issues by following the results of the test.