Hypothesis testing with normal distribution

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Yazomi
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for the two tailed test now do you know whether p(x>mean) or p(x<mean)
With x =sample mean
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MindMax2000
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Did not really understand the question. Anyway...

2 tailed tests: you're checking whether x-mean =0, so you're testing whether the range of data is within the a set confidence interval (usually 95%) e.g. 2.5% on the low end and 97.5% on the high end.

1 tailed tests: you're checking whether x>mean or x<mean. If x>mean, you're testing if the data is less than 95% mark. If x<mean, you're testing if the data is more than the 5% mark.

If the above does not clarify your query, please reply.
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Yazomi
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(Original post by MindMax2000)
Did not really understand the question. Anyway...

2 tailed tests: you're checking whether x-mean =0, so you're testing whether the range of data is within the a set confidence interval (usually 95%) e.g. 2.5% on the low end and 97.5% on the high end.

1 tailed tests: you're checking whether x>mean or x<mean. If x>mean, you're testing if the data is less than 95% mark. If x<mean, you're testing if the data is more than the 5% mark.

If the above does not clarify your query, please reply.
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Ahaa managed to figure out how to get a picture

So like for this question the writing in black is what’s given in the question and I just have to test the hypothesis at the stated level of significance.

For this,why do you find p(x>21.2) rather than x<21.2.

And when you test it against the sig level if either sides (left and right) are 0.025/2.5% then why would you choose to test it on the left side
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RDKGames
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(Original post by Yazomi)
Ahaa managed to figure out how to get a picture

So like for this question the writing in black is what’s given in the question and I just have to test the hypothesis at the stated level of significance.

For this,why do you find p(x>21.2) rather than x<21.2.

And when you test it against the sig level if either sides (left and right) are 0.025/2.5% then why would you choose to test it on the left side
21.2 is greater than the mean so you need to test it in the upper tail.

Otherwise P(x<21.2) will give you a probability greater than 0.5 hence you always fail to reject H0
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Theloniouss
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You are essentially taking a shortcut with this method. In reality, what you should be doing is calculating your critical values (i.e. the points on either tail where p(x<critical value) = significance level/2) and then checking whether the result is in the critical region.

With a two-tailed test, you can use your method on both tails but it doesn't make sense because you know the area on one side of your value will be the 'opposite' of the area on the other side.
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Yazomi
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(Original post by RDKGames)
21.2 is greater than the mean so you need to test it in the upper tail.
Omgggggg thank you so much!! That makes a lot more sense now
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Yazomi
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(Original post by Theloniouss)
You are essentially taking a shortcut with this method. In reality, what you should be doing is calculating your critical values (i.e. the points on either tail where p(x<critical value) = significance level/2) and then checking whether the result is in the critical region.

With a two-tailed test, you can use your method on both tails but it doesn't make sense because you know the area on one side of your value will be the 'opposite' of the area on the other side.
When you say p(x<critical value) how would you know whether to reject or not reject H0 because on the left side of the graph, x would be in the critical region but if it’s on the right side of the graph from the mean, x wouldn’t be in the critical region?
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Theloniouss
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(Original post by Yazomi)
When you say p(x<critical value) how would you know whether to reject or not reject H0 because on the left side of the graph, x would be in the critical region but if it’s on the right side of the graph from the mean, x wouldn’t be in the critical region?
sorry, you're right. It's p(x<critical value) for the left side and p(x>critical value) for the right side.
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MindMax2000
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(Original post by Yazomi)
When you say p(x<critical value) how would you know whether to reject or not reject H0 because on the left side of the graph, x would be in the critical region but if it’s on the right side of the graph from the mean, x wouldn’t be in the critical region?
I wouldn't quite put it that way, but yeah it's the right idea.

The aim of hypothesis testing is to see if you can reject the null hypothesis i.e. resulting values are outside of the confidence interval of the mean. If the x values are outside of the confidence interval, you can reject the null. In this case, presuming your calculations are correct, x is outside of the mean's confidence interval range, so you can say you have evidence to suggest x is different from the mean.
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Yazomi
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(Original post by Theloniouss)
sorry, you're right. It's p(x<critical value) for the left side and p(x>critical value) for the right side.
If that’s the case then how would you know whether to check the x value against the left side or the right side🤔
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Theloniouss
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(Original post by Yazomi)
If that’s the case then how would you know whether to check the x value against the left side or the right side🤔
You check it against both sides. Your critical region would be something like:

if x<a or x>b, reject H0
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Yazomi
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(Original post by MindMax2000)
I wouldn't quite put it that way, but yeah it's the right idea.

The aim of hypothesis testing is to see if you can reject the null hypothesis i.e. resulting values are outside of the confidence interval of the mean. If the x values are outside of the confidence interval, you can reject the null. In this case, presuming your calculations are correct, x is outside of the mean's confidence interval range, so you can say you have evidence to suggest x is different from the mean.
I’m currently trying to force myself to understand this😂😂 my brain is saying no to this
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