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# Hypothesis testing question (edexcel S2) Watch

1. sorry, i originally put this on the wrong forum:

I dont get what is going on here:

John wished to see if a coin was unbiased towards coming down heads. He decides that the level of significance of his test will be 5%.

If the random variable X represents the number of heads, what values of X would cause the null hypothesis to be rejected?

H0: p = 0.5 (the coin is not biased)

H1: p>0.5 (the coin is biased towards heads)

The probability of getting 7 or 8 heads is

3.1% + 0.4% = 3.5% (these values have been calculated using binomial
distribution X-B(8,0.5)

This is less than 5% and either X=7 or X=8 would cause the null hypothesis to be rejected.

Right, so this is my question:

Surely, if the probabiity of getting 7 or 8 heads is less than 5%, then it would lead to the null hypothesis NOT being rejected?

Thanks for your help.
3. (Original post by malteser12345)
Right, so this is my question:

Surely, if the probabiity of getting 7 or 8 heads is less than 5%, then it would lead to the null hypothesis NOT being rejected?

Thanks for your help.
The null hypothesis is that the coin is NOT biased.
If there is enough evidence that the coin is biased (i.e. X=7 or 8), then you reject this null hypothesis.

Think of it this way: You always start out by assuming the null hypothesis is true - e.g. "the coin is unbiased", or "there is no correlation in the data", or "it is a fair die", or "the average life expectancy is the same", etc.
You then look for evidence against the null hypothesis (to support the alternate hypothesis) - e.g. The coin landing on heads a lot, or significant correlation in the data, or the die landing on 6 a lot, or people living longer, etc.
You then reject the null hypothesis if the evidence you found is compelling enough - e.g. less than 5% chance of it being true, assuming the null hypothesis is true.

Make sense?
4. (Original post by tomthecool)
The null hypothesis is that the coin is NOT biased.
If there is enough evidence that the coin is biased (i.e. X=7 or 8), then you reject this null hypothesis.

Think of it this way: You always start out by assuming the null hypothesis is true - e.g. "the coin is unbiased", or "there is no correlation in the data", or "it is a fair die", or "the average life expectancy is the same", etc.
You then look for evidence against the null hypothesis (to support the alternate hypothesis) - e.g. The coin landing on heads a lot, or significant correlation in the data, or the die landing on 6 a lot, or people living longer, etc.
You then reject the null hypothesis if the evidence you found is compelling enough - e.g. less than 5% chance of it being true, assuming the null hypothesis is true.

Make sense?
Thanks for your help. However, if the probability of getting 7 or 8 heads is less than 5% (the level of significance), then the evidence surely suggests that it is unlikely that the coin is biased towards heads and the null hypothesis should not be rejected?

sorry if i am missing something obvious here!
5. (Original post by tomthecool)
The null hypothesis is that the coin is NOT biased.
If there is enough evidence that the coin is biased (i.e. X=7 or 8), then you reject this null hypothesis.

Think of it this way: You always start out by assuming the null hypothesis is true - e.g. "the coin is unbiased", or "there is no correlation in the data", or "it is a fair die", or "the average life expectancy is the same", etc.
You then look for evidence against the null hypothesis (to support the alternate hypothesis) - e.g. The coin landing on heads a lot, or significant correlation in the data, or the die landing on 6 a lot, or people living longer, etc.
You then reject the null hypothesis if the evidence you found is compelling enough - e.g. less than 5% chance of it being true, assuming the null hypothesis is true.

Make sense?
actually, tbh, my question is, what does the level of significance really mean? Because I think this is what i am misunderstanding?

Thanks.
6. (Original post by malteser12345)
Thanks for your help. However, if the probability of getting 7 or 8 heads is less than 5% (the level of significance), then the evidence surely suggests that it is unlikely that the coin is biased towards heads and the null hypothesis should not be rejected?

sorry if i am missing something obvious here!
Oh... no, you've misunderstood slightly.

It's saying "Suppose you toss the coin 8 times, and it lands on heads 7 or 8 times. THEN, you would reject the null hypothesis (and conclude that the coin is biased)".

The level of significance, in this case 5%, is saying "If the observed outcome had less than a 5% chance of happening, then this is significant enough for us to reach a new conclusion from".

Remember that you cannot ever prove anything with statistics! If I gave you a coin, which you flip 100 times and get 100 heads, would you know for certain that the coin is biased? No. But you would know with 99.999...% certainty that it is.
But we don't normally care about being 99.999% sure of something in statistics. Being just 99%, or even 95%, sure is usually good enough.
So when you test under the 5% significance level, you are saying "If I reject the null hypothesis, there is at most a 5% chance that I was wrong to do so".
7. (Original post by tomthecool)
Oh... no, you've misunderstood slightly.

It's saying "Suppose you toss the coin 8 times, and it lands on heads 7 or 8 times. THEN, you would reject the null hypothesis (and conclude that the coin is biased).
so, when would you not reject the null hypothesis? I was under the impression that, if the probability of getting 7 or 8 heads was under 5%, you would not reject the null hypothesis, and if it was over 5%, you would reject it.
8. (Original post by malteser12345)
so, when would you not reject the null hypothesis? I was under the impression that, if the probability of getting 7 or 8 heads was under 5%, you would not reject the null hypothesis, and if it was over 5%, you would reject it.
No, think about it: If you reject the null hypothesis in favour of the alternate hypothesis, then you are concluding that the coin is biased.
And if you get lots of heads (i.e. 7 or 8), you are clearly going to conclude that the coin is biased!

If the probability of getting 7 or 8 heads is <5%, and this event happens, then you are concluding that there must be something wrong with what you have assumed earlier - i.e. that the coin is unbiased. So you reject this hypothesis.
9. (Original post by tomthecool)
No, think about it: If you reject the null hypothesis in favour of the alternate hypothesis, then you are concluding that the coin is biased.
And if you get lots of heads (i.e. 7 or 8), you are clearly going to conclude that the coin is biased!

If the probability of getting 7 or 8 heads is <5%, and this event happens, then you are concluding that there must be something wrong with what you have assumed earlier - i.e. that the coin is unbiased. So you reject this hypothesis.
ok, so i get that. But also part of the question which i didnt put in was:

The probability of X=6 or greater is 14.4%, which is greater than the significance level so X=6 would NOT cause the hypothesis level to be rejected.

THIS is the bit im not sure about. So, 14.4% chance of 6 heads, which is greater than 5%, so you would conclude that the coin is boased towards coming down heads, and you would reject the null hypotheses?

Sorry if i am being a nuisance! I just dont get this!!
10. (Original post by malteser12345)
ok, so i get that. But also part of the question which i didnt put in was:

The probability of X=6 or greater is 14.4%, which is greater than the significance level so X=6 would NOT cause the hypothesis level to be rejected.

THIS is the bit im not sure about. So, 14.4% chance of 6 heads, which is greater than 5%, so you would conclude that the coin is boased towards coming down heads, and you would reject the null hypotheses?

Sorry if i am being a nuisance! I just dont get this!!
You reject the null hypothesis if the probability is LESS THAN the significance level, not more than!

From what you're saying...
Suppose you toss the coin 7 times and get 4 heads.
Assuming the coin is unbiased, the probability that you would have got 4 or more heads is 50%.
This is >5%... So would you conclude that the coin is biased if it landed on heads 4/7 times?!
Clearly you shouldn't - there's not enough evidence to support that conclusion!
11. (Original post by tomthecool)
You reject the null hypothesis if the probability is LESS THAN the significance level, not more than!
BUT WHY? What does the significance level really mean? I'm not even sure what it even means!!!!
12. (Original post by malteser12345)
BUT WHY? What does the significance level really mean? I'm not even sure what it even means!!!!
Let's say I have a hypothesis that the average weight of people is 70kg and that, given 10 people at random, there is less than a 20% chance that their average weight will be over 80kg. I want to test the idea that the average weight is actually more than 70kg.
Now I take 10 random people. Their average weight turns out to be 83kg.
So using a 20% significance level, I can say that my hypothesis that the average weight of a person being 70kg was wrong. Instead, the average is higher than that.
BUT if I tried to use a 5% significance level on this test, then I probably could not reach this same conclusion - because there would have been more than a 5% chance that the average weight of 10 random people is over 83kg!

Let's say I have a hypothesis that toast lands butter-side down 50% of the time. I want to test the idea that the probability is actually more than 50%.
I drop 1000 slices of toast, and 990 of them land butter side down. The probability that the bread would land like this (at least) this many times, assuming it was a 50-50 chance each time, is 0.00014 [I'm making this probability up; let's just pretend it's true rather than actually working it out :P]
So even using a 0.01% significance level, I can still reach the (very strong!) conclusion that my original hypothesis was wrong. Instead, the toast normally lands butter side down.

...Now let's go back to the original problem.
At a 5% significance level, which values of X are you allowed to use to conclude that the coin is biased? 7 and 8, because there is less than a 5% chance that the coin would land on heads this many times if the null hypothesis were true.
At a 1% significance level, which values of X are you allowed to use to conclude that the coin is biased? Only 8, because there is less than a 1% chance that the coin would land on heads this many times if the null hypothesis were true. But there is MORE than a 1% chance that the coin would land on heads 7 (or more) times, so you CANNOT reject the null hypothesis if X=7, at this significance level.
13. (Original post by tomthecool)
Let's say I have a hypothesis that the average weight of people is 70kg and that, given 10 people at random, there is less than a 20% chance that their average weight will be over 80kg. I want to test the idea that the average weight is actually more than 70kg.
Now I take 10 random people. Their average weight turns out to be 83kg.
So using a 20% significance level, I can say that my hypothesis that the average weight of a person being 70kg was wrong. Instead, the average is higher than that.
BUT if I tried to use a 5% significance level on this test, then I probably could not reach this same conclusion - because there would have been more than a 5% chance that the average weight of 10 random people is over 83kg!

Let's say I have a hypothesis that toast lands butter-side down 50% of the time. I want to test the idea that the probability is actually more than 50%.
I drop 1000 slices of toast, and 990 of them land butter side down. The probability that the bread would land like this (at least) this many times, assuming it was a 50-50 chance each time, is 0.00014 [I'm making this probability up; let's just pretend it's true rather than actually working it out :P]
So even using a 0.01% significance level, I can still reach the (very strong!) conclusion that my original hypothesis was wrong. Instead, the toast normally lands butter side down.

...Now let's go back to the original problem.
At a 5% significance level, which values of X are you allowed to use to conclude that the coin is biased? 7 and 8, because there is less than a 5% chance that the coin would land on heads this many times if the null hypothesis were true.
At a 1% significance level, which values of X are you allowed to use to conclude that the coin is biased? Only 8, because there is less than a 1% chance that the coin would land on heads this many times if the null hypothesis were true. But there is MORE than a 1% chance that the coin would land on heads 7 (or more) times, so you CANNOT reject the null hypothesis if X=7, at this significance level.
I think im starting to understand this now! Thanks for all your time and help, it is much appreciated.

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