# Hypothesis testing - Winning a game

• MHB
• mathmari
In summary, the conversation discusses formulating null and alternative hypotheses for a squash game between two players, A and B. It is determined that the second method of framing the hypotheses is preferred and the test statistic is given by the formula Z=(p_A-p_B)/sqrt(p(1-p)((1/n_A)+(1/n_B))). It is also clarified that only one sample is being considered, so a different formula for a 1-sample proportion test is needed. The null hypothesis is formulated as p=1/2 and the alternative hypothesis as p≠1/2. The conversation ends with the participants realizing that n=100 can already be filled in for the test statistic formula.
mathmari
Gold Member
MHB
Hey!

A and B play 100 games of squash; A wins E times. B claims that both of them have the same probability of winning a game.
We consider that the games are independent.

(a) Formulate the null hypothesis and the alternative hypothesis.
(b) For which values of $E$ will the null hypothesis be rejected with $\alpha=1\%$ and for which with $\alpha=5\%$ ?Is at (a) the null hypothesis $H_0: \ p=\frac{1}{2}$ and the alterinative hypothesis $H_1: \ p\neq \frac{1}{2}$ ? Or do we call $p_A$ the probability of $A$ and $p_B$ the probability of $B$, and so the null hypothesis is $H_0: \ p_A=p_B$ and the alterinative hypothesis $H_1: \ p\_A\neq p_B$ ? (Wondering)

Hey mathmari,

I would use the second way to frame this.

This link should be useful for comparing two samples of proportions.

Jameson said:
I would use the second way to frame this.

This link should be useful for comparing two samples of proportions.

So, we have the following:
$$H_0: \ p_A-p_B=0 \ \ \text{ and } \ \ H_1: \ p_A-p_B\neq 0$$

The test statistic is given by the formula $$Z=\frac{\hat{p}_A-\hat{p}_B}{\sqrt{\hat{p}(1-\hat{p})\left (\frac{1}{n_A}+\frac{1}{n_B}\right )}}$$

Is the size of the first population equal to the size of the second one, equal to $100$, i.e. $n_A=n_B=100$ ?

If yes, then we have that $\hat{p}_A=\frac{E}{100}$ and $\hat{p}_B=\frac{100-E}{100}$, or not?

Does it then hold that $\hat{p}=\hat{p}_A+\hat{p}_B=\frac{E}{100}+\frac{100-E}{100}=1$ ?

(Wondering)

mathmari said:
So, we have the following:
$$H_0: \ p_A-p_B=0 \ \ \text{ and } \ \ H_1: \ p_A-p_B\neq 0$$

The test statistic is given by the formula $$Z=\frac{\hat{p}_A-\hat{p}_B}{\sqrt{\hat{p}(1-\hat{p})\left (\frac{1}{n_A}+\frac{1}{n_B}\right )}}$$

Is the size of the first population equal to the size of the second one, equal to $100$, i.e. $n_A=n_B=100$ ?

If yes, then we have that $\hat{p}_A=\frac{E}{100}$ and $\hat{p}_B=\frac{100-E}{100}$, or not?

Does it then hold that $\hat{p}=\hat{p}_A+\hat{p}_B=\frac{E}{100}+\frac{100-E}{100}=1$ ?

Hey mathmari!

We only have 1 sample with $n=100$, so we can't do a 2-sample test of independent samples can we? (Wondering)

I like Serena said:
We only have 1 sample with $n=100$, so we can't do a 2-sample test of independent samples can we? (Wondering)

Oh yes. So we cannot apply the formula of the above link, can we? (Wondering)

mathmari said:
Oh yes. So we cannot apply the formula of the above link, can we?

Indeed. We need a different formula for a 1-sample proportion test.
It also means that we should have a hypothesis about a single proportion. (Thinking)

I like Serena said:
Indeed. We need a different formula for a 1-sample proportion test.
It also means that we should have a hypothesis about a single proportion. (Thinking)

So do we have the null hypothesis $H_0: p=\frac{1}{2}$ and the alternative hypothesis is $p\neq \frac{1}{2}$, or how do we formulate these hypotheses?

If we have these hypotheses, we get the following:

The test statistic is $$Z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}=\frac{\frac{E}{100}-\frac{1}{2}}{\sqrt{\frac{\frac{1}{2}\left (1-\frac{1}{2}\right )}{n}}}=\frac{\frac{E}{100}-\frac{1}{2}}{\sqrt{\frac{1}{4n}}}=\frac{\frac{E-50}{100}}{\frac{1}{2\sqrt{n}}}=\frac{(E-50)\sqrt{n}}{50}$$

Now we have to compare this with $z_{1-\alpha/2}$, or not? (Wondering)

All correct.
And we can already fill in n=100. Can't we? (Wondering)

I like Serena said:
All correct.
And we can already fill in n=100. Can't we? (Wondering)

Oh yes, you're right! Thank you so much! (Mmm)

mathmari said:
Oh yes. So we cannot apply the formula of the above link, can we? (Wondering)

Sorry mathmari! I quickly read the question and saw that you were formulating a two sample question but I didn't catch that it could be condensed into a one sample question. I'm glad ILS caught that and you solved the problem.

Jameson said:
Sorry mathmari! I quickly read the question and saw that you were formulating a two sample question but I didn't catch that it could be condensed into a one sample question. I'm glad ILS caught that and you solved the problem.

No problem! Thank you! (Smile)

## What is hypothesis testing in the context of winning a game?

Hypothesis testing is a statistical method used to determine whether the results of a study or experiment support a specific hypothesis. In the context of winning a game, it involves testing whether a particular strategy, technique, or factor has a significant impact on the likelihood of winning.

## How is a hypothesis formulated in the context of winning a game?

A hypothesis in the context of winning a game is typically formulated by identifying a specific aspect of the game that is believed to affect the outcome, and then stating a clear and testable prediction about how that aspect will impact the chances of winning. For example, a hypothesis could be "Using a certain strategy will significantly increase the chances of winning this game."

## What is the process of conducting a hypothesis test in the context of winning a game?

The process of conducting a hypothesis test in the context of winning a game typically involves collecting data on the strategy or factor being tested, and then using statistical analysis to determine whether the results support or reject the hypothesis. This can include calculating probabilities, comparing means or proportions, and other statistical techniques.

## What are some common mistakes to avoid when conducting a hypothesis test for winning a game?

Some common mistakes to avoid when conducting a hypothesis test for winning a game include not having a clear and testable hypothesis, failing to collect enough data, using inappropriate statistical tests, and misinterpreting the results. It is important to carefully plan and execute the test to ensure accurate and reliable results.

## How can hypothesis testing in the context of winning a game benefit players?

Hypothesis testing in the context of winning a game can benefit players by providing evidence-based insights into which strategies, techniques, or factors are most likely to contribute to winning. This can help players make more informed decisions and improve their chances of success in the game. Additionally, hypothesis testing can also lead to a better understanding of the game and its mechanics.

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