A tale about p-value

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What the p-value tells him is this – if it is assumed that the potion had no effect on the average fighting time of the couples, then what is the probability of picking out a sample that has an average fighting time of just 30 hours a week or less than that?

If, even without the effect of the potion, the probability of getting a value of 30 or less is very high, then it might be the case that the potion had no effect on the couples.

Assuming that all of us are familiar with the concept of a normal distribution, if the number of hours spent by couples fighting in a week is normally distributed with a mean of 32 hours, it would look like the graph  >>>.

Clearly, this means that there are couples lying on either side of the average, i.e. couples who fight a lot more than 32 hours a week, as well as couples who fight a lot less than 32 hours a week. So it is entirely possible to get a lower average due to chance selection of a sample that anyway fights less.

The doctor’s task is to find out that if this is what the distribution looks like, what is the probability of picking out couples who fight for 30 hours a week or less even without the effect of the potion.

How should the doctor calculate this value?

He frames his hypothesis:

Ho: Potion has no effect on no. of hours of fighting between couples

Ha: Potion reduces the no. of hours of fighting between couples

To start with, he assumes that the potion has no effect. If this is true, then what is the probability of getting an average fighting time = 30 hours a week (value obtained from the sample).

Ho:    μ  =  32

Ha:    μ   < 32

Let sample s.d. = 5 hours
Then estimated s.d. of population = 5/sqrt(64) = 0.625          (64 here is the size of sample)
Then the z-value associated with this is
(32-30)/0.625 = 3.2

Next he looks up the z-table to find the probability of getting this z value for a 1-tailed test, and gets a value of 0.000687. This is the value (p-value) he’s looking for.

What does this value tell the doctor?

This p-value of 0.000687 tells the doctor that if his potion in fact did not have any impact on the fighting time, there would only be a 0.000687 probability of getting an average fighting time of 30 hours or less for the sample selected.

Since this value is very very small, it would be reasonable for him to believe that the data does not support the null hypothesis.

So he can now start celebrating!!
His potion actually worked!
In a few days, he’ll surely be
rolling  in money!

What if he had got a higher p-value?

• Suppose the p-value obtained was 0.04. This would mean that there is a 4 in 100 chance of getting a sample with mean 30 even if the potion hasn’t worked.

• Similarly, suppose he gets a p-value of 0.1, then there’s a 10 in 100 chance (in other words, a 1 in 10 chance) of getting a sample with mean 30 even if the potion hasn’t worked.

• Knowing that there’s a 4 in 100 chance, or a 10 in 100 chance of picking out a sample that has a mean of 30 fighting hours  even without the effect of the potion, the doctor needs to decide whether he wants to take this as evidence that his potion has worked, or whether he thinks it is only chance that he picked out such a sample.

How to calculate p-value in general ?

1. Frame your hypothesis
2. Assume the null hypothesis to be true
3. Calculate the z or t value for getting the value in the alternative hypothesis
4. From the z/t-table, find the probability associated with the z or t value obtained above
5. This is the p-value you need to find

Use of p-value

• Once you’ve calculated the p-value, you know the probability of getting the sample value purely by chance.
• You decide on the type I error you’re willing to accept, i.e. the probability of rejecting a true null hypothesis that you find acceptable.
• Compare the p-value with the α-value chosen above.
• If α-value is less than the p-value, don’t reject the null hypothesis, else reject the null hypothesis.
• This is so because if the α-value is less than the p-value, it implies that in rejecting the null hypothesis, you’re willing to be wrong with a probability of only α, while given the sample, you would be wrong with a higher probability, p. This is an unacceptable situation.

Why is 0.05 usually taken as a cut-off for the p-value?

• In most studies, people are comfortable with the idea of being wrong in rejecting the null hypothesis 1 in 20 times, but not more than that. Hence the cut-off of 0.05(5/100 = 1 in 20 times type I error).

• However, this value is entirely subjective, & a higher or lower value can be chosen depending on the criticality of the study.

A quick recap

• The p-value, or probability value, tells us the probability of getting a value as small or as large as the one observed in the sample, given that our null hypothesis is true.

• In other words, we’re trying to find the likelihood of getting a value as small or as large as the sample statistic just by chance, and not as an outcome of the effect being studied.

• P-values are used to find the significance of results in hypotheses testing.

• For doing this, either some standard value of p(like 0.05) is considered, else, the p-value is compared with the α-value.

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