Question

a. What is hypothesis testing in statistics? Discuss

b. Does Type I error being considered more serious than Type II error? Explain

c. What is the p-value of a test? Give an example

Answer 1

Answers of A,B,C

A)
Hypothesis testing is a statistical method that is used in making statistical decisions using experimental data. Hypothesis Testing is basically an assumption that we make about the population parameter.Hypothesis testing was introduced by Ronald Fisher, Jerzy Neyman, Karl Pearson and Pearson’s son, Egon Pearson.  
The main purpose of statistics is to test a hypothesis. For example, you might run an experiment and find that a certain drug is effective at treating headaches. But if you can’t repeat that experiment, no one will take your results seriously. A good example of this was the cold fusion discovery, which petered into obscurity because no one was able to duplicate the results
Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis. Hypothesis testing is used to infer the result of a hypothesis performed on sample data from a larger population.
Key terms and concepts:
Null hypothesis: Null hypothesis is a statistical hypothesis that assumes that the observation is due to a chance factor. Null hypothesis is denoted by; H0: μ1 = μ2, which shows that there is no difference between the two population means.
Alternative hypothesis: Contrary to the null hypothesis, the alternative hypothesis shows that observations are the result of a real effect.
Level of significance: Refers to the degree of significance in which we accept or reject the null-hypothesis. 100% accuracy is not possible for accepting or rejecting a hypothesis, so we therefore select a level of significance that is usually 5%.
Type I error: When we reject the null hypothesis, although that hypothesis was true. Type I error is denoted by alpha. In hypothesis testing, the normal curve that shows the critical region is called the alpha region.
Type II errors: When we accept the null hypothesis but it is false. Type II errors are denoted by beta. In Hypothesis testing, the normal curve that shows the acceptance region is called the beta region.
Power: Usually known as the probability of correctly accepting the null hypothesis. 1-beta is called power of the analysis.
One-tailed test: When the given statistical hypothesis is one value like H0: μ1 = μ2, it is called the one-tailed test.
Two-tailed test: When the given statistics hypothesis assumes a less than or greater than value, it is called the two-tailed test.
Statistical decision for hypothesis testing:
In statistical analysis, we have to make decisions about the hypothesis. These decisions include deciding if we should accept the null hypothesis or if we should reject the null hypothesis. Every test in hypothesis testing produces the significance value for that particular test. In Hypothesis testing, if the significance value of the test is greater than the predetermined significance level, then we accept the null hypothesis. If the significance value is less than the predetermined value, then we should reject the null hypothesis. For example, if we want to see the degree of relationship between two stock prices and the significance value of the correlation coefficient is greater than the predetermined significance level, then we can accept the null hypothesis and conclude that there was no relationship between the two stock prices. However, due to the chance factor, it shows a relationship between the variables.
B)

A Type I error, on the other hand, is an error in every sense of the word. A conclusion is drawn that the null hypothesis is false when, in fact, it is true. Therefore, Type I errors are generally considered more serious than Type II errors. The probability of a Type I error (α) is called the significance level and is set by the experimenter. There is a tradeoff between Type I and Type II errors. The more an experimenter protects himself or herself against Type I errors by choosing a low level, the greater the chance of a Type II error. Requiring very strong evidence to reject the null hypothesis makes it very unlikely that a true null hypothesis will be rejected. However, it increases the chance that a false null hypothesis will not be rejected, thus lowering power. The Type I error rate is almost always set at .05 or at .01, the latter being more conservative since it requires stronger evidence to reject the null hypothesis at the .01 level then at the .05 level.
Explanation:
When considering a test of a null hypothesis H0 ,a type 1 error is the decision to reject the null (say it is false), when in fact it is true; while a type 2 error is a decision to accept the null (or "fail to reject it"), when in fact it is false.
In a courtroom, the null hypothesis is that the defendant is innocent while the alternative hypothesis is the opposite conclusion, that the defendant is guilty. Therefore, a type 1 error in this context is the conclusion that the defendant is guilty when in fact the person is innocent; and a type 2 error in this context is the conclusion that the defendant is innocent (or at least that there's not enough evidence to convict), when in fact the person is guilty.
Both kinds of errors are "bad", though societies typically think of the first kind of error as worse. The second kind of error here is more "dangerous" to the society, as it could result in letting a violent criminal go free.
In business, either kind of error can be "bad" or "dangerous". For example, if your company makes cars and you have tried to make some aspect of a crash test come out better (safer), the null hypothesis would be that the change makes no improvement to safety (or even makes safety worse) while the alternative hypothesis would be that the change does make an improvement to safety. In this context, a type 1 error would be the mistaken belief that the change has improved safety, when in fact it has not (so this could lead to more people dying and perhaps lawsuits). And a type 2 error would be the mistaken belief that the change has made no improvement when in fact it has (so this would mean a "missed opportunity" to improve safety, which could also lead to more people dying than they would otherwise).
C)
A p value is used in hypothesis testing to help you support or reject the null hypothesis. The p value is the evidence against a null hypothesis. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.
P values are expressed as decimals although it may be easier to understand what they are if you convert them to a percentage. For example, a p value of 0.0254 is 2.54%. This means there is a 2.54% chance your results could be random (i.e. happened by chance). That’s pretty tiny. On the other hand, a large p-value of .9(90%) means your results have a 90% probability of being completely random and not due to anything in your experiment. Therefore, the smaller the p-value, the more important (“significant”) your results.
When you run a hypothesis test, you compare the p value from your test to the alpha level you selected when you ran the test. Alpha levels can also be written as percentages.

P-values are the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the truth of the null hypothesis.
The p-value is calculated using the sampling distribution of the test statistic under the null hypothesis, the sample data, and the type of test being done (lower-tailed test, upper-tailed test, or two-sided test).

The p-value for:
a lower-tailed test is specified by: p-value = P(TS ts | H0 is true) = cdf(ts)
an upper-tailed test is specified by: p-value = P(TS ts | H0 is true) = 1 - cdf(ts)
assuming that the distribution of the test statistic under H0 is symmetric about 0, a two-sided test is specified by: p-value = 2 * P(TS |ts| | H0 is true) = 2 * (1 - cdf(|ts|))
Where:
P Probability of an event
TS Test statistic
ts observed value of the test statistic calculated from your sample
cdf() Cumulative distribution function of the distribution of the test statistic (TS) under the null hypothesis

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