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The chi-square goodness-of-fit test is a statistical method used to determine if a sample data matches a population with a specific distribution. In essence, it helps us verify claims about the distribution of categorical data. The method involves comparing the observed counts from the sample to the expected counts based on hypothesized proportions. This test is particularly useful when you want to see if there are significant differences between what was expected and what was actually observed.

• It quantifies the discrepancies between expected and observed data.
• It is applicable when you have one categorical variable with two or more levels.

The chi-square statistic is calculated using the formula: \[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\]where \( O_i \) represents the observed frequency, and \( E_i \) represents the expected frequency. After computing, the statistic is compared to a critical value from a chi-square distribution to make inferences.

Expected counts are the theoretically predicted frequencies of occurrences in each category of a sample, based on the null hypothesis. To calculate expected counts, you multiply the total number of observations by the proportion expected under the null hypothesis. This gives us an idea of what the distribution of the sample should look like if the null hypothesis is true.

• Calculation for each category: \( E = n \times p \), where \( n \) is the total sample size, and \( p \) is the proportion claimed.

For example, if a company claims that 52% of its mixed nuts are cashews, in a sample of 150 nuts, you would expect:

• Cashews: 150 \( \times \) 0.52 = 78
• Almonds: 150 \( \times \) 0.27 = 40.5
• Macadamia nuts: 150 \( \times \) 0.13 = 19.5
• Brazil nuts: 150 \( \times \) 0.08 = 12

These figures serve as benchmarks to compare against the actual sampled counts.

Proportions represent the relative part of the whole, often expressed as percentages or decimals. In the context of hypothesis testing, they are the expected percentages that a particular characteristic should take according to the null hypothesis.
In our problem, the company's claim sets forth proportions for each type of nut:

• 52% cashews
• 27% almonds
• 13% macadamia nuts
• 8% brazil nuts

These proportions form the basis of the null hypothesis. Understanding proportions is crucial because it guides how we calculate expected counts and analyze our sample data. When actual sample data varies considerably from these proportions, it might suggest that the claim doesn't hold, prompting a chi-square test.

In statistical testing, formulating the null and alternative hypotheses is essential, as they define what you are testing. The null hypothesis (H_0) is a statement of no effect or no difference, suggesting that any deviation in the data is due to chance.

• For our problem, the null hypothesis is that the actual proportions match the company's stated values: \( H_0: p_{cashews}=0.52, p_{almonds}=0.27, p_{macadamia}=0.13, p_{brazil}=0.08 \).

Contrastingly, the alternative hypothesis (H_a) suggests that there's some effect or difference, meaning at least one of the true proportions differs from the stated values.

• In our testing, the alternative hypothesis claims that the proportions do not match the company's claims entirely or partially.

Testing these hypotheses through observed data helps determine if there's enough evidence to reject the null hypothesis. This framework is central to statistical inference, allowing us to make sound decisions or conclusions based on sample data.