For the conjecture "The average rent of an apartment is more than $850 per month," what is the alternative hypothesis?
Scientists seek to establish theories or discover laws that explain observations or the results of experiments. The first step is to construct a hypothesis, or attempted explanation, for a set of facts, and then to test it. Usually, statistical methods are used: a sample of data is examined to see if it supports the proposed explanation. Typically, a null hypothesis, which contradicts the explanation will be constructed — this is normally denoted by H_{0} — while the explanation itself is called the alternative hypothesis, denoted by H_{A}. It is initially assumed that H_{0} is true, and the task of the researcher is to show that the data do not support this conclusion.
Usually, H_{0} and H_{A} are two mutually exclusive statements — they cannot both be true. They should also be exhaustive; that is, they should cover all the possible outcomes of the experimental investigation. A sample of data is obtained, against which the null hypothesis will be tested. The sample must be of sufficient size to enable valid conclusions to be drawn, and must be free from any bias that might affect the result.
The researchers must then establish a value, or one or or more sets of values, that would not support H_{0}. If the data are found to be in agreement with these values, the null hypothesis will be rejected, and the alternative hypothesis can then be said to be probably true. The test data can often be represented as a graph, with a peak in the middle and a “tail” on either side. Typically, most of the values for the thing being tested will cluster around the middle of the range, tailing off toward the low and high extremes. For example, a set of measurements of the heights of a large sample of people will show the majority around the middle of the range, and smaller numbers toward the very short and very tall ends.
There are three types of tests that can be applied to a set of data. In a right-tailed test, it has been determined that data that are above a certain value, known as the critical value, do not support the null hypothesis; in a left-tailed test, these data lie below the critical value; in a two-tailed test, the data that do not support H_{0} lie above and below a certain value or range of values. It is not possible to completely disprove the null hypothesis; instead, researchers must agree on an interpretation of the data based on how likely it is that H_{0} would be rejected when it is actually true. This likelihood is known as the significance level. For example, if a certain proportion of the data are above the critical value in a right-tailed test, this might indicate that there is only a 1% chance that H_{0} is true.
A drug company may be testing the results of a new treatment to reduce cholesterol. In this case, the null hypothesis would be that cholesterol levels are not reduced after taking the drug, while the alternative hypothesis would be that levels decrease. H_{0} would be assumed to be true, and researchers would then gather data to be analyzed in an attempt to reject it.
The data might consist of cholesterol measurements in a sample of people before and after taking the drug, compared with a similar sample who did not take it, over the same period. The researchers might then agree on how much of a reduction, and in what proportion of the sample who took the drug, can be regarded as significant. This information can be used to set a critical value, such as a reduction of 10% in 80% of those who took the drug. If the data fall above these values, the null hypothesis is rejected, and the alternative hypothesis accepted.
anon331576
Post 3 |
For the conjecture "The average rent of an apartment is more than $850 per month," what is the alternative hypothesis? |
FernValley
Post 2 |
@recapitulate, I like this example, although it can actually be easily proven, though most scientific situations of a null vs. alternative hypothesis cannot. However, you have a point that many of us use this concept in our lives without realizing it. |
recapitulate
Post 1 |
The concept of the null hypothesis vs. alternative hypothesis can be useful to anyone, not just scientists. When a person drives somewhere every day, for example, he or she often tries to prive that the route taken is the fastest way to get there. In this scenario, the null hypothesis is that another route, perhaps one suggested by a friend, is not faster. The alternative hypothesis is that the route taken is fastest. |