# Type 1 and type 2 errors in statistics pdf and cdf

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*The time to event or survival time usually follows certain skewed probability distributions.*

## Type I and Type II Errors

We now give some examples of how to use the binomial distribution to perform one-sided and two-sided hypothesis testing. Determine whether the die is biased. We use the following null and alternative hypotheses:. Example 2 : We suspect that a coin is biased towards heads.

When we toss the coin 9 times, how many heads need to come up before we are confident that the coin is biased towards heads? INV 9,. DIST 7, 9,. DIST 6, 9,. They have now changed their manufacturing process and hope that this has improved the reliability. To test this, they took a sample of 24 components produced using the new process and found that 13 components passed the quality assurance test.

Does this show a significant improvement over the old process? We use a one-tailed test with null and alternative hypotheses:. DIST 12, 24,. Example 4 : It is commonly believed that drivers of flashy-colored cars red, yellow, pink, orange or purple get pulled over more often for a driving violation. It is possible, however, that drivers of these cars are pulled over no more often or even less often. To get a deeper insight into this issue, a study was made of the 50 cars that were pulled over in one month and it was found that 7 cars had a flashy color.

Determine whether flashy colored cars are pulled over differently from any other colored car. INV 50,. Since 7 is between these values, we cannot reject the null hypothesis and so there is no evidence that the police are pulling over drivers of flashy cars more or less often than drivers of other cars. DIST 7,50,. If of the 50 cases, 4 had been for flashy cars, then we would have rejected the null hypothesis since 4 is less than the left-side critical value.

DIST 4,50,. Note that there is a lack of symmetry here since. DIST 5,50,. DIST 15,50,. A random sample of 20 bottles finds that 6 of these sampled bottles are defective. Which test based on the binomial distribution would you use to answer this question?

Please help me with this question: Assume we randomly tested 10 individuals living in the rural area, and found that only 3 of them were positive for Zika virus infection.

AK, Glad that you like the post. I have just added a fourth example, which is a two-tailed test. In this case we are not able to reject H0, but what is the p-value? DIST 5, 24, 0. This shows that it is highly unlikely that the process is an improvment. Great post. Is it right to say that it is easier to get a sig.

Hi Bob, I am not sure what you mean by where the comparison is occurring. By definition a significant result can only occur at the tails, but I am not sure that this is what you are asking. My test of expected pass rate as follows: 1. Two outcome failed, pass 3. Probability of failure 1. INV Tosses,0. DIST I get p values above 0. Hello Bruce, This is problem is similar to Example 2 on this webpage.

The problem is likely to be that the last argument in your formula is 0. It is a nice description of how to perform a one-sided test for Binomial data.

I wonder if there exist a practical recommendation for how to do this in a two-sided case. Because the strong theory says we should look for an unbiased most powerful test, but I could not find any reference to a practical implementation of the respective procedure. Andrey, You can treat Example 3 as a two-tailed test. An example of this is winners in a lottery. Too few and people are not motivated to play; two many and the company loses money.

Thank you Charles. H1:p not equal to 0. Andrey, Yes, you can test this null hypothesis. The approach is similar. Thank you very much, Charles. I wanted to use the McNemar test but apparently it is recommended to use a binomial test or sign test?

Could you tell me if this is correct and if yes, should I do a two-tailed test? Also, how do I run a binomial test when the answer is yes or no and not a percentage? Thank you in advance for your help,. For the first and third examples, you use one less than the number of successes mentioned.

DIST function. What is your reasoning for doing this? Caroline, In the first example, you want to find out the probability that three comes up 4 of more times i. These examples calculate a two tailed confidence interval. You need to use the one-tailed critical value instead of the two-tailed critical value. The other side of the confidence interval is infinity or negative infinity depending on whether you using the right or left critical value Charles.

The number of credit card holders of a bank in two different cities city — X and city — Y settling their excess withdrawal amounts in time without attracting interest follows binomial distribution. The manager collections of the bank feels that the proportion of the number of such credit card holders in the city — X is not different from the proportion of the number of such credit card holders in the city — Y.

Similarly a sample of credit card holders is taken from the city — Y and it is found that 50 of them are settling their excess withdrawal amount in — time without attracting interest, check the intuition of the sales manager at a significance level of 0. In any case, whether or not this is a homework assignment, here is a hint: Look at the two sample hypothesis testing for the Proportion Distribution at Proportion Distribution Charles.

As my understanding, p-value is the probability that, using a given statistical model, the statistical summary such as the sample mean difference between two compared groups would be the same as or more extreme than the actual observed results Wikipedia , given the null hypothesis is true. Similarly, Example 2: dbinom 6,9,. Same conclusion, but weaker. William, Thank you for catching these errors. I have now corrected the referenced webpage. On behalf of all the users of this website, I appreciate your help in improving the accuracy and quality of the website.

I am a statistic student with a question. With hypothesis test proportion binomial distribution, is it possible to have a left tail? DIST 13, 24,. I am still analyzing the subject and found that, for example, Mathematica and Maple return values equal to those of Excel. Statistical software in general associates the inverse of the distribution function F x to quantiles, calculate using the criterion of the BINOM. INV function. In what concerns the code sent, I think there is one situation in which no correction should be made to the Excel value:.

CRIT function, where sometimes the values are different. INV function, we must take into account the following: 1. The function does not follow the rules you presented for an inversion function. This fact is reflected when saying that alpha is the criterion value and not significance level or type I error.

The function does not even know if the user is considering the right or left tail. In view of this: 1 the right tail c. To perform this correction it is necessary to indicate to which tail if the value wanted refers.

This can be achieved at spreadsheet level using formulas or with UDF function such as the one below:. Thanks Antonio for the clear explanation.

I will add this function to the Real Statistics Resource Pack to help people with this concept. I found this site very helpful. Thank you. So the way I see it, to be confident at a minimum level of alpha, I could only reject if I observed greater than or equal to 8 heads.

Mike, Thanks for your comment. Please let me know whether you agree with his approach. To look to see if the rate in a given country is significantly different from the overall worldwide average rate is it valid us use BINOMDIST no of cases, number in sample group, worldwide average rate, TRUE and look to see if the value is 0.

## Normal distribution

In probability theory , a normal or Gaussian or Gauss or Laplace—Gauss distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. It states that, under some conditions, the average of many samples observations of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors , often have distributions that are nearly normal. Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies.

In null hypothesis significance testing , the p -value [note 1] is the probability of obtaining test results at least as extreme as the results actually observed , under the assumption that the null hypothesis is correct. Reporting p -values of statistical tests is common practice in academic publications of many quantitative fields. Since the precise meaning of p -value is hard to grasp, misuse is widespread and has been a major topic in metascience. If we state one hypothesis only and the aim of the statistical test is to see whether this hypothesis is tenable, but not, at the same time, to investigate other hypotheses, then such a test is called a significance test. In essence, a claim is assumed valid if its counterclaim is highly implausible. Thus, the only hypothesis that needs to be specified in this test and which embodies the counterclaim is referred to as the null hypothesis ; that is, the hypothesis to be nullified.

In this tutorial, we discuss many, but certainly not all, features of scipy. The intention here is to provide a user with a working knowledge of this package. We refer to the reference manual for further details. There are two general distribution classes that have been implemented for encapsulating continuous random variables and discrete random variables. Over 80 continuous random variables RVs and 10 discrete random variables have been implemented using these classes. Besides this, new routines and distributions can be easily added by the end user. If you create one, please contribute it.

## Hypothesis Testing for Binomial Distribution

We now give some examples of how to use the binomial distribution to perform one-sided and two-sided hypothesis testing. Determine whether the die is biased. We use the following null and alternative hypotheses:.

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Introduction to power in significance tests.