[Statistics Notes] (7) Hypothesis testing

(7) Hypothesis testing

one question

A bodybuilding club whose main goal is to lose weight claims that participating in its training courses can at least reduce the average weight loss of the person who loses weight by more than 8.5kg. In order to verify the credibility of the claim, the investigators randomly selected 10 participants and obtained their weight records as follows:

At $\alpha =0.05$ the level of significance, do the findings support the club ’s claims?

How can we answer this question?

Consider the above table, another form of expression is as follows (sample difference calculation table):

The following is the mean and standard deviation of the difference:

From this we have come to the conclusion that there is evidence that the club ’s claims are credible.

After reading the book for a long time, what is this? This requires starting with the basic idea of hypothesis testing.

Statistical language uses an equation or inequality to express the original hypothesis of the problem.

The $H_{0}$ original hypothesis used in this question is $H_{0}$ :: $\ mu _ {0} - \ mu _ {1} \ geq 8.5$ ; Then the $H_{1}$ alternative hypothesis is used. That is, if the null hypothesis is not established, the null hypothesis must be rejected and a choice must be made in another hypothesis. This hypothesis is called the alternative hypothesis. As shown in this $H_{1}$ example: $\ mu _ {0}-\ mu _ {1} <8.5$ : .

The null hypothesis and the alternative hypothesis are mutually exclusive. To affirm the null hypothesis means to abandon the alternative hypothesis; to reject the null hypothesis means to accept the alternative hypothesis.

$\ mu$ Is the parameter we want to check.

$\alpha$ Is the significance test level, in this question: $\alpha =0.05$

Two types of errors ( $\alpha$ errors and $\beta$ errors)

The propositions proposed for the null hypothesis can be expressed by the null hypothesis or the null hypothesis. Of course, we make judgments based on the information provided by the sample, that is, the department infers the population. Therefore, the judgment may be correct or incorrect, that is, we are faced with the possibility of making mistakes. There are two types of mistakes:

The first type of error: the original hypothesis $H_{0}$ is true, but it was rejected by us. The probability of making such an error is $\alpha$ expressed in terms of it, so it is also called $\alpha$ an error or a true error.

The second type of error: the original hypothesis $H_{0}$ is false, but it has not been rejected by us. The probability of making this error is $\beta$ expressed in terms of it, so it is also called $\beta$ an error or a false error.

When the null hypothesis $H_{0}$ is true, we reject it, and the probability of making this mistake is $\alpha$ expressed as, then, when it $H_{0}$ is true, we have not rejected it $H_{0}$ , indicating that the correct decision has been made, and the probability is naturally $1-\alpha$ ; when the null hypothesis $H_{0}$ is false However, we did not refuse $H_{0}$ . The probability of making such a mistake is $\beta$ expressed in terms of, then, when it $H_{0}$ is false, we refuse $H_{0}$ , which is also the correct decision, and its probability is $1-\beta$ . Summarized as follows:

project	No rejection $H_{0}$	Refuse $H_{0}$
$H_{0}$ True	$1-\alpha$ (Correct decision)	$\alpha$ (Abandon true error)
$H_{0}$ False	$\beta$ (Take false error)	$1-\beta$ (Correct decision)

In hypothesis testing, we are in the implementation of such a principle, that is, first control commit $\alpha$ an error in principle.

Illustration of two types of mistakes made in hypothesis testing.

Steps for hypothesis testing:

propose assumption
Determine appropriate test statistics
Prescribed significance level $\alpha$
Calculate the value of the test statistic
Make statistical decisions

Make statistical decisions:

Calculate test statistics
According to the given significance level $\alpha$ , look up the table to get the corresponding critical value $z_{\alpha }$ or $z_{\alpha /2}$ , $t _ {\ alpha}$ or $t _ {\ alpha / 2}$
Compare the value of the test statistic with $\alpha$ the critical value of the level
Draw a conclusion about accepting or rejecting the null hypothesis

Left unilateral test and right unilateral test

Left side inspection (lower limit inspection)

Right one-sided inspection (upper limit inspection)

Two-sided test

hypothetical test

Hypothesis testing (Hypothesis Testing), also known as statistical hypothesis testing, is a statistical inference method used to determine whether the sample-to-sample, sample-to-population difference is caused by sampling error or essential difference. Significance testing is the most commonly used method in hypothesis testing, and it is also the most basic form of statistical inference. Its basic principle is to first make some assumptions about the characteristics of the population, and then make statistical assumptions through sampling research. Should be rejected or accepted to make an inference.

The basic idea of hypothesis testing is the principle of "small probability event", and its statistical inference method is a method of reproof with some kind of probabilistic nature. The idea of small probability means that a small probability event will basically not happen in an experiment. The idea of the counter-proof method is to first propose a test hypothesis, and then use appropriate statistical methods, using the principle of small probability, to determine whether the hypothesis is true. In order to test whether a hypothesis $H_{0}$ is correct, first assume that the hypothesis is $H_{0}$ correct, and then $H_{0}$ make a decision to accept or reject the hypothesis according to the sample . If the sample observations lead to "small probability events", the hypothesis should be rejected $H_{0}$ , otherwise the hypothesis should be accepted $H_{0}$ .

The so-called "small probability event" in hypothesis testing is not an absolute contradiction in logic, but is based on the principle widely adopted by people in practice, that is, a small probability event hardly occurs in an experiment, but how small is the probability Counting as a "small probability event", obviously, the smaller the probability of the "small probability event", the $H_{0}$ more convincing it is to reject the null hypothesis. It is often remembered that the probability value is α (0 <α <1), which is called the significance of the test Level. For different problems, the significance level α of the test is not necessarily the same. It is generally believed that the probability of an event occurring is less than 0.1, 0.05, or 0.01, that is, a "small probability event".

Commonly used hypothesis testing methods include Z test, t test, chi-square test, F test, etc.

The basic principle of hypothesis testing

Using the t distribution and the residual set of interval events in interval estimation are small probability events and the principle of small probability , the value of the test statistic t and the rejection domain are obtained. When the sample is representative, the statistic t and the rejection domain are available Make better decisions for testing. This method is called t-test and the product quality test is replaced by a large sample by a small sample.

The basic steps

1, also known as the null hypothesis is proposed to test the hypothesis, the symbol is $H_{0}$ ; the alternative hypothesis symbol is $H_{1}$ .
$H_{0}$ : Sample-to-population or sample-to-sample differences are caused by sampling errors;:
$H_{1}$ sample-to-population or sample-to-sample differences exist; the
pre-set test level is 0.05; when the test hypothesis is true, but is wrong The probability of rejection is denoted as α, which is usually taken as α = 0.05 or α = 0.01.
2. Select the statistical method, and calculate the size of the statistic from the sample observation value according to the corresponding formula, such as X2 value and t value.

According to the type and characteristics of the data, Z test, T test, rank sum test and chi-square test can be selected respectively.
3. According to the size and distribution of the statistics, determine the probability P of the test hypothesis and establish the result.

If P> α, the conclusion is that the level taken by α is not significant and does not refuse $H_{0}$ , that is, the difference is likely to be caused by sampling error and is not statistically valid;

If P ≤ α, the conclusion is that according to the level of α taken is significant, rejected $H_{0}$ , accepted $H_{1}$ , it is considered that this difference is unlikely to be caused only by sampling error, it may be caused by different experimental factors, so it is statistically true. The size of the P value can generally be obtained by consulting the corresponding boundary value table.

Issues that need attention:

Before making hypothesis tests, you should pay attention to whether the data itself is comparable.
When the difference is statistically significant, it should be noted whether such difference is meaningful in practical applications.
Select the correct hypothesis testing method according to the type and characteristics of the data.
Determine whether to choose one-sided inspection or two-sided inspection based on professional and experience.
Judgment conclusions cannot be absolute. It should be noted that no matter accepting or refusing to test hypotheses, there is a possibility of judgment errors.

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