Proving Power Function for H0: p=1/2 in Coin Bias Test | X=8, 9, or 10

In summary, a test was conducted to determine if a coin was biased towards heads, with the hypothesis H0: p=1/2 and H1: p>1/2. The test involved counting the number of heads, X, in 10 tosses of the coin and rejecting H0 if X=8, 9, or 10. The power function for this test is given by p^8(45-80p+36p^2), which expresses the probability of obtaining X > 7 in 10 throws using the binomial formula.
  • #1
sara_87
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A coin is suspected of bias towards heads, so a test is made of the hypothesis H0: p=1/2 against H1:p>1/2, where p is the probability of heads. The test is to count the number of heads, X, in 10 tosses of the coin, and reject H0 if X=8, 9, or 10.
Show that the power function for this test is given by: p^8(45-80p+36p^2)

I have no idea how to start. I know that the power function means: P(reject H0 given H0 is false) but i don't know how to continue.

Any help would be very much appreciated.
Thank you
 
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  • #2
I think you can ignore the part "given H0 is false" because the test is expressed for an arbitrary p (which may or may not be the value of p under H1).

So, all you need is to express the probability of obtaining X > 7 in 10 throws using the binomial formula.
 
  • #3
Oh right
Thank you v much!
(I should have known that)
 

FAQ: Proving Power Function for H0: p=1/2 in Coin Bias Test | X=8, 9, or 10

1. What is the significance of proving the power function for H0: p=1/2 in a coin bias test?

The power function in a hypothesis test measures the ability of the test to reject a false null hypothesis. In the case of a coin bias test, proving the power function for H0: p=1/2 allows us to determine the probability of correctly identifying a biased coin (i.e. one that does not have an equal chance of landing on heads or tails) as unbiased.

2. How is the power function calculated in a coin bias test?

The power function in a coin bias test can be calculated using the binomial distribution formula, which takes into account the sample size, the probability of success (in this case, p=1/2 for an unbiased coin), and the significance level of the test.

3. What sample sizes are typically used in a coin bias test?

The sample size used in a coin bias test can vary, but it is typically recommended to have a minimum of 30 coin flips to ensure the results are statistically significant. However, larger sample sizes may be used to increase the power of the test.

4. How do results of the coin bias test impact the interpretation of the power function?

If the results of the coin bias test show that the null hypothesis is rejected and the coin is identified as biased, this indicates that the power function has successfully identified the true state of the coin. However, if the test fails to reject the null hypothesis and the coin is identified as unbiased, this may indicate a lower power function and a higher chance of incorrectly identifying a biased coin as unbiased.

5. Are there any limitations to using the power function in a coin bias test?

Like any statistical test, the power function in a coin bias test has its limitations. It assumes that the sample is representative of the population and that the coin flips are independent of each other. Additionally, the power function does not guarantee a correct identification of a biased coin, but rather provides a probability of correctly identifying it.

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