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A basketball game is played for 30 minutes. A coach claims that his team's players commit, on average, no more than 10 fouls per game. Let ยต represent the team's average number of fouls per game. Another coach thinks that these players create more fouls. And in the next game the team fouled 100 times.

Q. Can I utilize the hypothesis testing without applying CLT?

Q. Can I directly use the probability of having such an event or even rarer events to say that the probability of occurrence of such a event or further rarer event is below 1% for Poisson distribution, with the given parameters (10 fouls per game per 30 minutes), hence the claim of the team's coach may be rejected.

Q. If not, can you justify why? Suggest links and theory behind it to make understanding.

Kapil
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1 Answers1

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  1. Yes we can use hypothesis testing without CLT i.e without the normality assumption, hypothesis testing is a general notion to test between validity of different assertions.There are parametric and non parametric ways to do this, in a parametric way we assume that our data $X$ follows some kind of distribution with a certain parameter i.e $X$~$F_\mu$ where $F$ is a distribution function and $\mu$ the parameter that exactly defines the distribution, then we can do hypothesis testing on this parameter or functions of this parameter by deciding upon a test rule to reject or accept the assertion on the parameter based on some probabilistic measure on the available data, one common way to do this is the p value method where we check for probabilities of extreme observations under our assertion for the parameter , and reject our assertion if this probability is very low.
  2. Assuming that the data follows a poisson distribution, we can define the null hypothesis $H_0:\mu=10$ vs $H_1:\mu>10$ i.e the first coaches assumption that they create 10 fouls on average vs the other coaches observation that they create more fouls,since our only available observation is $X=100$ we find the p value , $P(X\geq 100|\mu=10)$ which is very close to 0, implying that observing something like this under our assumption is nearly impossible, which suggests that our assumption must be wrong and thus we can reject the null hypothesis and conclude that
  3. For this problem the assumption of a poisson distributed random variable sounds valid, but maybe this is kind of contextual, in a professional match one would expect a large number of actions by players and low number of fouls which is valid for a poisson assumption but in non professional matches these numbers might not be valid, beside we are making our results based on only one observation which is not always suitable as standard statistical theory always reccomends larger sample sizes to make important inferences.

Sources :

  1. Chapter 9 Neyman-Pearson Theory of Hypothesis testing, Introduction to Probability and Statistics by V.K.Rohatgi and Saleh.
  2. https://en.wikipedia.org/wiki/Poisson_distribution
hopeless.cantor
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