For The Purposes Of This Problem, Assume The Roulette Has 52

For The Purposes Of This Problem Assume the Roulette Has 52 Numbers

For the purposes of this problem, assume the roulette has 52 numbers, 26 red and 26 green (there are no 0s). You want to create an experiment where you get to watch the roulette N times, and observe the numbers that it picks (from 1–52, red and green). Your null hypothesis is that the roulette is “fair,” i.e., it picks green and red numbers with equal probability. The alternative hypothesis is that it is not fair, and it picks red with a probability of p = 0.50 + δ, where δ is a small number (positive or negative, note that under the null δ = 0). Assuming a significance level of 1%, how large does N have to be to have a test with power of 90% against the alternative that δ = 0.005? Assuming a significance level of 1%, how large does N have to be to have a test with power of 90% against the alternative that δ = 0.002?

Paper For Above instruction

The problem explores the sample size determination necessary to achieve a specified statistical power in testing the fairness of a roulette with 52 numbers, 26 red and 26 green, under a binomial model. The null hypothesis (H₀) posits that the roulette is fair, with an equal probability of 0.5 for red or green, while the alternative hypothesis (H₁) suggests a deviation from fairness, with the probability of red being p = 0.5 + δ. The focus is on determining the minimal number of observations N required to detect a small deviation δ with 90% power at 1% significance level, for two different values of δ: 0.005 and 0.002.

Statistical Framework

The problem involves hypothesis testing for a binomial proportion. Let X denote the number of times the roulette lands on red in N trials. Under H₀, X follows a binomial distribution with parameters N and p = 0.5, i.e., X ~ Binomial(N, 0.5). The test is formulated to reject H₀ if the observed proportion deviates sufficiently from 0.5, indicating possible unfairness.

The significance level α = 0.01 corresponds to the probability of a Type I error (rejecting the null hypothesis when it is true). The test employs a critical value derived from the binomial distribution or, more conveniently, from its normal approximation due to the large N assumption.

Normal Approximation

For large N, the binomial distribution approximates a normal distribution with mean μ = Np and variance σ² = Np(1 - p). Under H₀, the standardized test statistic (z-score) is given by:

z = (X - N0.5) / sqrt(N 0.25)

where 0.25 = 0.5 * 0.5. The critical z-value corresponding to a one-sided test at α = 0.01 is approximately 2.33, since:

• For a right-sided test, rejection occurs if z > z_α = 2.33.

Calculating Sample Size N

To achieve the desired power (1 - β = 0.9), the non-centrality parameter under the alternative must be sufficiently large to surpass the critical value with probability 90%. The test statistic under H₁ centers around Np with p = 0.5 + δ, so its standardized form (when considering the true p) is:

z_alt = (X - N0.5) / sqrt(N0.25)

For the test to have power 90%, the distribution under H₁ must exceed the critical value 2.33 (for the right tail) with probability 0.9. Equivalently, the non-central distribution shifted by the true p should satisfy:

• z_alt = z_critical + z_power = 2.33 + 1.28 ≈ 3.61

because z_power ≈ 1.28 is the z-score for 90% power.

Deriving N for a Given δ

Expressing the above, the equation becomes:

(N (0.5 + δ) - N 0.5) / sqrt(N * 0.25) = 3.61

which simplifies to:

δ N / (0.5 sqrt(N)) = 3.61

or:

δ * sqrt(N) / 0.5 = 3.61

thus:

sqrt(N) = (3.61 * 0.5) / δ = 1.805 / δ

and finally:

N = (1.805 / δ)²

Application to Specified δ Values

1. For δ = 0.005:

N = (1.805 / 0.005)² = (361)² = 130,321

2. For δ = 0.002:

N = (1.805 / 0.002)² = (902.5)² ≈ 814,502

These calculations indicate that to detect a deviation of δ = 0.005 with 90% power at a 1% significance level, approximately 130,321 observations are needed. For an even smaller deviation of δ = 0.002, the required sample size increases to about 814,502 observations, reflecting the increased difficulty in detecting smaller effects.

Conclusion

In summary, the necessary sample size N to achieve desired power and significance level in testing the fairness of the roulette can be efficiently approximated using the normal approximation method under the binomial model. The relationship N ≈ (1.805 / δ)² emphasizes that detecting smaller differences in probability (δ) demands a significantly larger number of observations. Practically, these enormous sample sizes underscore the challenges associated with detecting minute deviations from fairness in probabilistic experiments and highlight the importance of precision and power analysis in experimental design.

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