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A little exercise. Let's suppose we are working in company which tell us : our bulbs light have a average duration of 800 hours.

So our null hipotesys is 800 hours.

We make aleatory test of 50 light bulbs and we get:

$$ \bar x = 750 \text{ hours}\\ s = 120\text{ hours}\\ n = 50\\ H_0 = 800\text{ hours}\\ \dfrac{750 - 800}{\frac{120}{\sqrt{50}}} \approx - 2.91 $$

When we looking for in the Z score table we get 2.91 have a probability distribution of 0.0018.

This would be equivalent to 0.18%. So we say we have 0.18% of probability to make type I error, and we say we have very little probability to make a mistake.

Until here all is right. But How to explain this result to low level people ?

QUESTION IS : Is this the same as saying : " We have $99.82\%$ $(100-0.18) $of probability the average of bulbs lights is $750$ hours or less "

Dave
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NIN
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  • Use pictures! For example, a number line with the null distribution and empirical distribution plotted together would help your audience. – DifferentialPleiometry Nov 04 '21 at 18:54
  • "So we say we have 0.18% of probability to make type I error, and we say we have very little probability to make a mistake." This is false. – Dave Nov 04 '21 at 19:05
  • @Dave ok. Is false. What is the interpretation ? – NIN Nov 04 '21 at 19:09
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    What's the definition of a p-value? What's the definition of $\alpha?$ // I edited your post to format the math in text instead of a picture. I was tempted to make a mathematical correction, but I think it is best for you to think about the mistake and correct it. Hint: it involves your null hypothesis. Do you see what the mistake is? – Dave Nov 04 '21 at 19:12
  • @Dave . That's why I don't put the significanse level. I want to directly say the probability rather than say : " bellow this level ( for example 5%) our alternative hypotesis is fulfilled" – NIN Nov 04 '21 at 19:24
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    Then what is the definition of a p-value? // Did you figure out the mistake with your null hypothesis? // I invite you to have the fun of considering how you would respond to a client who says, "You got 750, which is 800-ish. What's the problem?" Even more fun is if you had gotten $\bar x = 799$, particularly if the sample size is gigantic. – Dave Nov 04 '21 at 19:26
  • @NIN You require a confidence level for this test, if that is what you mean by "significance level". If you otherwise mean the $p$-value, you should also report that value. – DifferentialPleiometry Nov 04 '21 at 19:30
  • @Dave You are quite right. Custumers say : 750, 760 ,740.. . Whats the problem ? it's almost the same. . . . So in my exercise I am making the P value right ?. . . is it 0.18 the p-value? ? Not to me, it's just probability of make tipe I error. . ohh its so confusing – NIN Nov 04 '21 at 19:53
  • Sorry I didn't figure out the mistake in null hypotesis. . The null hypotesis is just the company which make the light bulbs say : the light bulbs last 800 hours on. Period. – NIN Nov 04 '21 at 19:57
  • Have *what* as 800 hours? – Dave Nov 04 '21 at 19:58
  • Relevant [answer](https://stats.stackexchange.com/a/72583/7071). – dimitriy Nov 04 '21 at 19:58
  • @Dave . I mean : bulbs last 800 hours on – NIN Nov 04 '21 at 20:00
  • So what would be the mathematical way to write the null hypothesis that bulbs last 800 hours on average? – Dave Nov 04 '21 at 20:20
  • @Dave . You are right . It is incorectly expressed. It is not equal ( = ). The null hypotesis is expressed with colon (:). H0 : 800 hours. – NIN Nov 04 '21 at 20:50
  • What is 800 hours? – Dave Nov 04 '21 at 21:04
  • @Dave . A average. OMG dave. You confusing me. – NIN Nov 05 '21 at 01:21
  • So then what, in English (we’ll get to the math), is the null hypothesis? – Dave Nov 05 '21 at 01:23
  • @Dave , the null hypotesis is the one which is established as true. – NIN Nov 05 '21 at 10:27
  • Establishes *what* as true? What is the null hypothesis in your case? – Dave Nov 05 '21 at 11:30
  • @Dave, I surrender. – NIN Nov 05 '21 at 21:18
  • $H_0:\mu=800$ (Why not $\bar x=800?$) – Dave Nov 05 '21 at 21:32
  • @Dave . Because is a sample average. – NIN Nov 06 '21 at 03:06
  • Exactly…you don’t have to use the sample to estimate something about the sample. What you don’t know is the population parameter. – Dave Nov 06 '21 at 03:45
  • @Dave . Ahhh ok. You're quite right. A little mistake. Thank you very much. – NIN Nov 06 '21 at 19:07

1 Answers1

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The result is hard to explain to non-technical people because it is hard to explain even to statisticians!

The biggest problem is that you are applying the wrong tool incorrectly to the problem that you wish to solve. For a 'hypothesis test' the probability of a false positive is set prior to the analysis as the alpha (or size of the test). It is what Galen is calling a "confidence level". A hypothesis test generates a decision regarding rejection of the null hypothesis but it does not tell you the probability of a light bulb failing early or late, but can be designed to protect you against exceeding the alpha level probability of sending out a batch of bulbs with a mean failure rate lower (or higher) than 800 hours. In other words it can be set up as a useful test for acceptance of the batch of light bulbs.

A hypothesis test does not tell you directly about the probability of a light bulb failure time.

The p-value of 0.0018 that you obtained is the result from a 'significance test', and it says that the observed test results argue fairly strongly against the null hypothesis. It does not tell you the probability of a false positive error because the decision to accept or reject the null hypothesis is not forced by the p-value and so it is dependent on the p-value in combination other considerations. When any decision to reject the null is optional and dependent on factors outside the statistical test procedure the rate of erroneous decisions is not a property of the statistical test. Therefore a significance test does not even have a type I error rate.

Confusion regarding hypothesis test and significance tests is widespread and there are many relevant questions on this site. You can begin here: What is the difference between "testing of hypothesis" and "test of significance"? and here: Interpretation of p-value in hypothesis testing

You might like to report the probability of a light bulb in this batch lasting substantially less (or more) than 800 hours. The tools you want to use for that would not be a hypothesis test or a significance test, but one of several methods that estimate the distribution of light bulb lifetimes. You might like to look at bootstrapping for a very interesting way to do that without having to assume a particular distributional shape.

Michael Lew
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  • waoooo, very clear. . .So in my example I just say : there are strong evidence that average of light bulbs is not 800 hours. . . . But I can't say more. – NIN Nov 04 '21 at 20:24
  • This would be even harder to explain to non technical people. – NIN Nov 04 '21 at 20:25
  • Your data would allow you to say more, but you're right about that test result in isolation. – Michael Lew Nov 04 '21 at 22:58