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Answer Q SticiGui : Hypothesis Testing: Does Chance explain the Results? Hypothesis Tests An experimenter suspects that a certain die is "loaded;" that is, the chances that the die lands on different faces are not all equal. Recall that dice are made with the sum of the numbers of spots on opposite sides equal to 7: 1 and 6 are opposite each other, 2 and 5 are opposite each other, and 3 and 4 are opposite each other. The experimenter decides to test the null hypothesis that the die is fair against the alternative hypothesis that it is not fair, using the following test. The die will be rolled 50 times, independently.

If the die lands with one spot showing 13 times or more, or 3 times or fewer, the null hypothesis will be rejected. Under the null hypothesis, the distribution of the number of times the die lands showing one spot is binomial with parameters n=50, p=1/6. Under the null hypothesis, the expected number of times the die lands showing one spot is 8.3333 and the standard error of the number of times the die lands showing one spot is √(50×(1/6)×(5/6)) ≈ 2.89. The significance level of this test is the probability of rejecting the null hypothesis when it is actually true, which corresponds to the probability that under the binomial distribution with n=50 and p=1/6, the number of times one spot appears is either 13 or more or 3 or fewer.

This involves calculating tail probabilities of the binomial distribution. Using normal approximation to the binomial (since n=50 is sufficiently large), with mean 8.33 and standard deviation approximately 2.89, the critical values are determined by standard normal z-scores corresponding to these counts.

For the upper tail, when the count is 13 or more:

• Calculate z = (13 - 8.33) / 2.89 ≈ 1.58, then find the area to the right of z=1.58, which is approximately 0.057.

Similarly for the lower tail, when the count is 3 or fewer:

• calculate z = (3 - 8.33) / 2.89 ≈ -1.85, area to the left of z=-1.85 ≈ 0.032.

Therefore, the significance level (probability of Type I error) for this test is approximately the sum of these tail probabilities: 0.057 + 0.032 ≈ 0.089, or 8.9%. This means there is about an 8.9% chance of wrongly rejecting the null hypothesis if the die is actually fair.

Paper For Above instruction

Hypothesis testing is a fundamental method in statistics used to make inferences about populations based on sample data. In the context of testing whether a die is fair or loaded, the null hypothesis posits that each face of the die has an equal probability of landing face up (p=1/6), while the alternative hypothesis suggests that the probabilities differ from this neutrality. This analysis involves a binomial distribution since each roll can be considered a Bernoulli trial with two outcomes: landing on a specific face or not.

The experiment involves rolling the die 50 times independently. The key statistic of interest is the number of times the face with one spot appears. Under the null hypothesis, the expectation (mean) of this count is n×p=50×(1/6)=8.3333. The variability (standard deviation) of this count is calculated by the standard binomial formula: √(n×p×(1−p)) = √(50×(1/6)×(5/6)) ≈ 2.89, which quantifies the spread of the distribution assuming the die is fair.

The decision rule for the hypothesis test is specified: reject the null hypothesis if the face of 'one' appears 13 or more times, or 3 or fewer times. These cutoff points correspond to the tail areas under the binomial distribution, and are associated with the significance level of the test—that is, the probability of obtaining such extreme results if the null hypothesis is true. Using a normal approximation, the z-scores corresponding to these cutoff points can be calculated, which facilitate estimation of the tail probabilities.

In practice, the z-score for the upper tail (13 or more appearances) is approximately 1.58, leading to an area to the right of about 0.057. For the lower tail (3 or fewer appearances), the z-score is approximately -1.85, with an area to the left of about 0.032. Summing these gives an overall significance level of roughly 8.9%. This indicates an about 8.9% chance of falsely rejecting the null hypothesis representing a fair die based on this test configuration.

Such hypothesis tests are vital in experimental statistics, quality control, and industrial processes where decision-making depends on probabilistic evidence. They provide a structured approach to assess whether observed deviations from expected outcomes are likely due to chance or indicative of systemic bias, such as a loaded die in this case. The method hinges on understanding the underlying distribution, choosing appropriate critical values, and properly interpreting the resulting p-values and significance levels.

References

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