Need help understanding when the null hypothesis should be rejected Watch

Illidan2
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'In a manufacturing process the proportion(p) or faulty articles has been found, from long experience, to be 0.1.

A sample of 100 articles from a new manufacturing process is tested, and 8 are found to be faulty.

The manufacturers wish to test at the 5% level of significance whether or not there has been a reduction in the proportion of faulty articles.'

a)Suggest a suitable test statistic.
b) Write down two suitable hypotheses.
c) Explain the condition under which the null hypothesis is rejected.

I have done parts a and b. I am trying to really understand thoroughly what is going on with hypothesis testing. I don't quite understand the condition under which the null hypothesis would be rejected. If someone could explain exactly what would happen here, I would really appreciate it
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CandidateZero
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If the probability you obtain from the information provided (forming the binomial distribution) is a significant result, which is to say that the probability is less than the significance level given, then your result is significant and in the critical region of values for which you must reject the null hypothesis. In your case, the significance level is 0.05, so any probability P you obtain such that P < 0.05 gives you ground to reject your null hypothesis for the alternative one.
(Original post by Illidan2)
'In a manufacturing process the proportion(p) or faulty articles has been found, from long experience, to be 0.1.

A sample of 100 articles from a new manufacturing process is tested, and 8 are found to be faulty.

The manufacturers wish to test at the 5% level of significance whether or not there has been a reduction in the proportion of faulty articles.'

a)Suggest a suitable test statistic.
b) Write down two suitable hypotheses.
c) Explain the condition under which the null hypothesis is rejected.

I have done parts a and b. I am trying to really understand what is going on with hypothesis testing. I don't quite understand the condition under which the null hypothesis would be rejected. If someone could explain exactly what would happen here, I would really appreciate it
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sotor
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okay so hypothesis testing is testing how likely a particular outcome is, if all the conditions are assumed to be true. so if the % that are faulty is 10% and that is still true, is getting 8% faulty possible simply due to chance.

this is because even though overall in the population (every piece ever made) 10% might be faulty (this is an average), if you take a random sample there could be more or less than 10% (variation around the mean). that's what i mean by happened due to chance

so if you test at the 5% level, youre testing whether p(X<8) is less than 5%. this means that if p(X<8) is less than 5%, it is considered unlikely to have happened due to chance. therefore, you would reject.
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Illidan2
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I think I understand somewhat. Thank you
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sotor
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(Original post by Illidan2)
I think I understand somewhat. Thank you
feel free to ask anything you dont get! i spent weeks trying to understand hypothesis testing last year, i feel your struggle
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Illidan2
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Thank you As it happens, I do need further clarity on another similar issue.

"A random variable has a distribution B(20,p). A single observation is used to test H0: p=0.15 against H1: p<0.15. Using a 5% level of significance, find the critical region of this test".

I used the binomial cumulative distribution and found that the probabilities less than 5%, or 0.05, occurred on two occasions:

x less than or equal to 0, and x greater than or equal to 6. I checked the solution afterwards, and found that only x less than or equal to zero was taken into consideration. Do we only take into consideration the probability for less than or equal to a particular value when the alternative hypothesis is that the probability is less than or equal to a particular value? If so, why is this the case?
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Illidan2
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(Original post by Illidan2)
Thank you As it happens, I do need further clarity on another similar issue.

"A random variable has a distribution B(20,p). A single observation is used to test H0: p=0.15 against H1: p<0.15. Using a 5% level of significance, find the critical region of this test".

I used the binomial cumulative distribution and found that the probabilities less than 5%, or 0.05, occurred on two occasions:

x less than or equal to 0, and x greater than or equal to 6. I checked the solution afterwards, and found that only x less than or equal to zero was taken into consideration. Do we only take into consideration the probability for less than or equal to a particular value when the alternative hypothesis is that the probability is less than or equal to a particular value? If so, why is this the case?
Based on further research it seems that what I did would be applicable if I was doing a two-tailed test, but a one-tailed test is named such because it only requires testing one end, or 'tail', of the distribution.
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sotor
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(Original post by Illidan2)
Do we only take into consideration the probability for less than or equal to a particular value when the alternative hypothesis is that the probability is less than or equal to a particular value? If so, why is this the case?
Yes this is correct. it's honestly complicated to understand why and I still don't know 100% but if H1: p< then you test (X<x).
when you do the normal distribution in y13 you'll be able to see why, but for now its hard to explain. the gist is that you only have to test values which are equal to or more extreme than H1, so if H1 is < then any values > are not more extreme than H1.
(Original post by Illidan2)
Based on further research it seems that what I did would be applicable if I was doing a two-tailed test, but a one-tailed test is named such because it only requires testing one end, or 'tail', of the distribution.
this is correct, but if it is a two tailed test then you need to half the probability. so if H1: p does not equal Y then you test if p(X<x) is less than 2.5%, not 5% (or if p(X>x) is less than 2.5%. again will make more sense when you learn the normal distribution but for now just learn it.

when you're learning about hypothesis testing with the binomial at AS, its mostly important to learn what the concept behind hypothesis testing is and learn the steps to carry it out. then next year you'll develop your understanding. for a reference for how complicated it is, only 2 teachers out of more than 8 in my school are able to understand it! you just have to practise and practise the questions until you can hit all the marking points without thinking.
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Illidan2
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(Original post by sotor)
Yes this is correct. it's honestly complicated to understand why and I still don't know 100% but if H1: p< then you test (X<x).
when you do the normal distribution in y13 you'll be able to see why, but for now its hard to explain. the gist is that you only have to test values which are equal to or more extreme than H1, so if H1 is < then any values > are not more extreme than H1.

this is correct, but if it is a two tailed test then you need to half the probability. so if H1: p does not equal Y then you test if p(X<x) is less than 2.5%, not 5% (or if p(X>x) is less than 2.5%. again will make more sense when you learn the normal distribution but for now just learn it.

when you're learning about hypothesis testing with the binomial at AS, its mostly important to learn what the concept behind hypothesis testing is and learn the steps to carry it out. then next year you'll develop your understanding. for a reference for how complicated it is, only 2 teachers out of more than 8 in my school are able to understand it! you just have to practise and practise the questions until you can hit all the marking points without thinking.
Well, let's hope I get there! I'm self-taught you see, to sit the exams for the full A-Level in June 2019. I do not have a teacher whatsoever so having you available to ask is a great help It probably seems odd that i'm doing Hypothesis testing with the binomial distribution approx 2 months before exams, but i've finished pure content for both years, and mechanics for both years. Almost all of the new spec stats content is heavily featured in year 1, with only a few new topics coming up in year 2. As such, once i've got this sorted, I essentially only have the normal distribution, exponential regression and a little bit of conditional probability to go through.
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sotor
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(Original post by Illidan2)
Well, let's hope I get there! I'm self-taught you see, to sit the exams for the full A-Level in June 2019. I do not have a teacher whatsoever so having you available to ask is a great help It probably seems odd that i'm doing Hypothesis testing with the binomial distribution approx 2 months before exams, but i've finished pure content for both years, and mechanics for both years. Almost all of the new spec stats content is heavily featured in year 1, with only a few new topics coming up in year 2. As such, once i've got this sorted, I essentially only have the normal distribution, exponential regression and a little bit of conditional probability to go through.
well done for finishing all of pure and mech! youre right, y13 stats is not that heavy so good luck

what exam board are you?
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Illidan2
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Edexcel. How about you? How did you manage with the A-Level? If you don't mind me asking, what grade did you end up with?


(Original post by sotor)
well done for finishing all of pure and mech! youre right, y13 stats is not that heavy so good luck

what exam board are you?
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(Original post by Illidan2)
Edexcel. How about you? How did you manage with the A-Level? If you don't mind me asking, what grade did you end up with?
im taking it this year! ive just finished all the stats already
i got an A at AS last year and im hoping for an A* this year but realistically might be an A, we'll see haha
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