After getting trounced by your little brother in a...

After getting trounced by your little brother in a children's game, you suspect that the die he gave you is unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the die's fairness? $$ \begin{array}{c|r} \text { Face } & \text { Count } \\ \hline 1 & 11 \\ 2 & 7 \\ 3 & 9 \\ 4 & 15 \\ 5 & 12 \\ 6 & 6 \end{array} $$ a) If the die is fair, how many times would you expect each face to show? b) To see if these results are unusual, will you test goodnessof-fit, homogeneity, or independence? c) State your hypotheses. d) Check the conditions. e) How many degrees of freedom are there? f) Find $\chi^2$ and the $\mathrm{P}$-value. g) State your conclusion.

After getting trounced by your little brother in a children's game, you suspect that the die he gave you is unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the die's fairness? $$
\begin{array}{c|r}
\text { Face } & \text { Count } \\
\hline 1 & 11 \\
2 & 7 \\
3 & 9 \\
4 & 15 \\
5 & 12 \\
6 & 6
\end{array}
$$
a) If the die is fair, how many times would you expect each face to show?
b) To see if these results are unusual, will you test goodnessof-fit, homogeneity, or independence?
c) State your hypotheses.
d) Check the conditions.
e) How many degrees of freedom are there?
f) Find $\chi^2$ and the $\mathrm{P}$-value.
g) State your conclusion.

Key Concepts

Expected Frequency

Expected frequency refers to the number of times an outcome is predicted to occur in a probability experiment under a given hypothesis or model. In the context of a fair die, it is determined by multiplying the total number of trials by the probability of the outcome, and it serves as the baseline for comparing observed data against expected outcomes.

Goodness-of-Fit Test

The goodness-of-fit test is a statistical procedure used to determine how well an observed frequency distribution matches an expected distribution. It helps assess whether deviations between observed and expected counts can be attributed to random chance or if they suggest a flaw in the assumed model.

Hypothesis Testing

Hypothesis testing involves setting up a null hypothesis, which posits no effect or no difference (e.g., the die is fair), and an alternative hypothesis, which claims that there is an effect or difference (e.g., the die is biased). This framework provides a systematic way to make decisions about the population based on sample data.

Conditions for the Chi-Square Test

In order to validly apply the chi-square test, certain conditions must be met, including independence of observations and sufficiently large expected counts (usually at least 5 for each category). These conditions ensure that the chi-square approximation to the true distribution is reliable.

Degrees of Freedom

Degrees of freedom in a chi-square goodness-of-fit test are determined by the number of categories minus one, potentially adjusted for any estimated parameters. They represent the number of independent pieces of information available to estimate variability in the data after fitting the model.

Chi-Square Statistic

The chi-square statistic is computed as the sum of the squared differences between observed and expected frequencies for each category, divided by the corresponding expected frequency. This statistic quantifies the overall discrepancy between the observed data and the model's predictions.

P-Value

The P-value is a measure used in hypothesis testing to assess the strength of evidence against the null hypothesis. It represents the probability of obtaining a chi-square statistic as extreme as, or more extreme than, the one computed from the data, assuming the null hypothesis is true.

Conclusion in Hypothesis Testing

Drawing a conclusion in hypothesis testing involves comparing the P-value to a pre-determined significance level (commonly 0.05). If the P-value is less than the significance level, the null hypothesis is rejected, indicating that the observed data provide sufficient evidence to support the alternative hypothesis.

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