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Hypothesis testing is a crucial part of statistics that allows us to make inferences or decisions about a population based on sample data. In our nut mixing example, we want to see if the machine actually mixes nuts in the expected ratio of 5:2:2:1.

• The **null hypothesis** ( H_0 ) states that there is no difference between the observed and expected mixes. H_0: The machine mixes the nuts in the ratio 5:2:2:1.
• The **alternative hypothesis** ( H_a ) proposes that the mix is not according to the expected ratio. H_a: The machine does not mix nuts in the ratio 5:2:2:1.

We use statistical tests, like the Chi-square test, to determine whether to reject the null hypothesis or not. Choosing to reject or not depends on the comparison between the critical value and the computed test statistic. Ensuring our inference is both rigorous and controlled by statistical measures is the aim of hypothesis testing.

Expected frequencies are calculated based on the hypothesis that is being tested. In the context of the nut mixing problem, they represent the quantities we would anticipate if the machine were functioning correctly, according to the proposed ratio of 5:2:2:1.
To calculate expected frequencies:

• **Peanuts:** 50% of 500 = 250
• **Hazelnuts:** 20% of 500 = 100
• **Cashews:** 20% of 500 = 100
• **Pecans:** 10% of 500 = 50

Each expected frequency derives from the total number, 500, and adheres strictly to the proportions outlined in the hypothesis. These frequencies are later compared to the observed frequencies using the Chi-square test to see if they align with the machine's output.

Observed frequencies are what we see directly in the data collected from our experiment or observation. These are the actual counts of the nuts provided by the machine. In this exercise, students will understand how real-world outputs are compared against theoretical or expected values.
For this problem:

• **Peanuts:** Observed = 269
• **Hazelnuts:** Observed = 112
• **Cashews:** Observed = 74
• **Pecans:** Observed = 45

The differences between observed and expected frequencies will be scrutinized in the Chi-square test. Observed frequencies are original data points, key in assessing if there’s a deviation from what was anticipated.

The level of significance plays a critical role in hypothesis testing as it defines the probability threshold for assessing evidence against the null hypothesis. Often denoted as α , it represents the risk we are willing to take of making a Type I error—rejecting a true null hypothesis.
In this nut mixing analysis, we use a 0.05 level of significance. This suggests that there is a 5% risk of asserting there is an issue with the nut mix, when there may not be. ‌Choosing a level of significance involves balancing rigor in detecting effects and safety against false alarms.
‌ Once this level is set, it is utilized to find the critical value in a Chi-square distribution table, helping guide the decision whether to reject the null hypothesis based on the test statistic calculated.