It is hypothesized that a CroOkeD Die is actually a fair die. Two crooked dice are pictured below.

You decide to perform goodness of fit test to decide whether a CroOkeD Die is fair or not. You decide to use a significance level of 0.1.

To perform this test, you need to collect some data. The data is included in the following video.

Based on this data, what is the value of the test statistic? (21:80)

Question

It is hypothesized that a CroOkeD Die is actually a fair die. Two crooked dice are pictured below.

You decide to perform goodness of fit test to decide whether a CroOkeD Die is fair or not. You decide to use a significance level of 0.1.

To perform this test, you need to collect some data. The data is included in the following video.

Based on this data, what is the value of the test statistic?  (21:80)

BlackTom AI · Accepted Answer

Before diving into analysis, note the prompt presents a goodness-of-fit test setup for a CroOked Die and asks for the test statistic based on data from a video. However, the provided data or observed counts are not included in the text you shared, and there are no answer options to evaluate.

Here is how I would approach this problem if the data were available:

- Step 1: Identify the null hypothesis and expected frequencies. H0: the die is fair, so each face 1–6 has probability 1/6. With a sample size n, the expected count for each face would be E_i = n/6.
- Step 2: Collect observed counts O_i from the data (the video data in this problem would give the number of times each face appeared).
- Step 3: Compute the chi-square statistic using the formula: χ² = Σ (O_i − E_i)² / E_i over i = 1 to 6. This measures the discrepancy between observed and expected frequencies under H0.
- Step 4: Determine degrees of freedom. For a die with 6 faces, df = number of categories − 1 = 5 (assuming no parameters estimated from the data; if you estimated a parameter from the data, df would be reduced accordingly).
- Step 5: Compare χ² to the critical value from the chi-square distribution at the chosen significance level α = 0.1, or compute a p-value to decide whether to reject H0.

Potential pitfalls and quick checks:
- If any O_i = 0, the term (O_i − E_i)² / E_i is still valid as long as E_i > 0; ensure you use the correct E_i = n/6 for all faces under H0.
- If the sample size n is small, the chi-square approximation may be unreliable; in that case, consider alternative methods or combine categories if necessary, though with a fair die each category should still be treated as 6 outcomes unless data constraints require merging.
- If the video data yields a non-integer or miscounted O_i, recalculate from scratch to avoid arithmetic errors.

Since no observed counts, sample size, or answer options are provided in the text, I cannot compute or verify a numeric test statistic like 5.04 here. If you share the observed counts for each face (or the raw data from the video) and the total n, I can walk through the exact calculation step by step and confirm the test statistic value and the subsequent decision at α = 0.1.

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