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In a Chi-Square Goodness-of-Fit Test, 'Expected Frequency' is a key term. It refers to the number that is predicted or expected for each category based on a specific probability distribution under the null hypothesis.
To find this, we multiply the total sample size by the proportion specified in the hypothesis for each category. In our smoking example, we have aged-based proportions like 0.25 for both the age groups \(<16\) and \( \geq21\). These proportions help us predict how many smokers we expect to fall into each age group if the null hypothesis is true.
So, if we have a total of 1031 smokers, the expected frequencies are calculated as follows:

• For age group \(<16\): \(257.75\) males \( (1031 \times 0.25=257.75) \).
• For age group \(16-17\): \(206.2\) males \( (1031 \times 0.2=206.2) \).
• For age group \(18-20\): \(309.3\) males \( (1031 \times 0.3=309.3) \).
• For age group \( \geq21\): \(257.75\) males, the same as the first group.

These expected frequencies provide a baseline to compare against the actually observed data, revealing how closely the observed data fits the expected distribution.

The null hypothesis in a Chi-Square Goodness-of-Fit Test is an essential logical statement. It posits that there is no significant difference between the observed frequencies and those expected by the predicted distribution of categories.
In simpler terms, it claims that any difference we see between what we expect and what we observe is due to random chance and not because the distribution is wrong.
In our cigarette smoking age exercise, the null hypothesis \( (H_0) \) is defined by the proportions \( p_1, p_2, p_3, \) and \( p_4 \). These proportions represent the assumption that the age groups \(<16\), \(16-17\), \(18-20\), and \( \geq21\) make up 25%, 20%, 30%, and 25% of the sample, respectively.
By establishing these proportions, the null hypothesis makes a clear statement: the actual ages at which people start smoking follow this predicted pattern. By testing this hypothesis, we can understand if our assumptions about these proportions are correct or if there's a significant deviation suggesting some other pattern.

When conducting a Chi-Square Goodness-of-Fit Test, 'Observed Frequency' is what you actually see in the data you are analyzing. It's the count of occurrences within each category from your samples. Observed frequencies are critical because they help you determine whether there is a significant difference from the expected frequencies.
In our example, the observed frequencies are the number of males falling into different age groups:\(<16, 16-17, 18-20,\) and \( \geq21\). We have:

• \(<16\): 237 males
• \(16-17\): 258 males
• \(18-20\): 320 males
• \(\geq21\): 216 males

With these numbers, we can structure the Chi-Square calculation, looking for variations between what's observed and what's expected under the null hypothesis. Any significant deviation could mean our original assumptions aren't holding up well to scrutiny.

Understanding the 'Degrees of Freedom' (df) is crucial when performing statistical tests like the Chi-Square Goodness-of-Fit test. Simply put, degrees of freedom refer to the number of values that are free to vary in the final calculation of statistics after certain restrictions are applied. For this test, it's calculated as the number of categories minus one.
In our smoking example, there are four age categories (\(<16, 16-17, 18-20, \geq21\)). Thus, the degrees of freedom is \(4-1=3\).
Why does this matter? The degrees of freedom are used to determine the critical value from the Chi-Square distribution table. This critical value sets the threshold to decide whether to reject the null hypothesis. In our test case, with 3 degrees of freedom at a 0.05 significance level, the critical value is approximately 7.81. Our calculated Chi-Square statistic must be compared against this threshold to determine the hypothesis's validity. Hence, understanding degrees of freedom is integral to making informed decisions based on your statistical test outcomes.