You are trying to determine if a certain die is fair (has a uniform distribution). You roll the die 60 times and record the outcomes in the table below. Find the test statistic, [tex]$\chi_0^2$[/tex], for the chi-square goodness-of-fit test. If necessary, round the final answer to one decimal place.
[tex]$\chi_0^2=\sum_k \frac{(O-E)^2}{E}$[/tex]
| Outcome | 1 | 2 | 3 | 4 | 5 | 6 |
| :------- | :-: | :-: | :-: | :-: | :-: | :-: |
| Expected | 10 | 10 | 10 | 10 | 10 | 10 |
| Observed | 14 | 5 | 6 | 7 | 10 | 18 |
Answer (2)
We calculated the chi-square test statistic for the die rolls using the formula χ 0 2 = ∑ k E ( O − E ) 2 . The contributions from each outcome were summed, resulting in a final statistic of χ 0 2 = 13.0 . This value is essential for determining if the die is fair or not.
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Calculate the chi-square statistic for each outcome using the formula E ( O − E ) 2 .
Sum the chi-square statistics for all outcomes to find the total chi-square test statistic, χ 0 2 .
The chi-square statistics for each outcome are: 1.6, 2.5, 1.6, 0.9, 0, 6.4.
The total chi-square test statistic is: χ 0 2 = 1.6 + 2.5 + 1.6 + 0.9 + 0 + 6.4 = 13.0 .
Explanation
Understand the problem and provided data We are given a problem where we need to find the chi-square test statistic, χ 0 2 , for a goodness-of-fit test. We have the observed and expected values for rolling a die 60 times. The formula for the chi-square test statistic is given as χ 0 2 = ∑ k E ( O − E ) 2 , where O is the observed value and E is the expected value.
Calculate chi-square statistic for each outcome We need to calculate the chi-square statistic for each outcome and then sum them up to find the total chi-square test statistic. Let's calculate the chi-square statistic for each outcome:
Outcome 1: 10 ( 14 − 10 ) 2 = 10 4 2 = 10 16 = 1.6 Outcome 2: 10 ( 5 − 10 ) 2 = 10 ( − 5 ) 2 = 10 25 = 2.5 Outcome 3: 10 ( 6 − 10 ) 2 = 10 ( − 4 ) 2 = 10 16 = 1.6 Outcome 4: 10 ( 7 − 10 ) 2 = 10 ( − 3 ) 2 = 10 9 = 0.9 Outcome 5: 10 ( 10 − 10 ) 2 = 10 0 2 = 10 0 = 0 Outcome 6: 10 ( 18 − 10 ) 2 = 10 8 2 = 10 64 = 6.4
Sum the chi-square statistics Now, we sum the chi-square statistics for all outcomes:
χ 0 2 = 1.6 + 2.5 + 1.6 + 0.9 + 0 + 6.4 = 13.0
State the final answer The chi-square test statistic is 13.0.
Examples
The chi-square goodness-of-fit test is used in various real-world scenarios. For example, it can be used to determine if the distribution of M&M colors in a bag matches the distribution claimed by the manufacturer. It can also be used in genetics to check if observed offspring ratios match expected Mendelian ratios. In marketing, it can be used to assess whether customer preferences for different product features are uniformly distributed or if certain features are more popular than others. In each of these cases, the chi-square test helps to determine if the observed data significantly deviates from the expected data, providing valuable insights for decision-making.
