91Ó°ÊÓ

In hypothesis testing, the null hypothesis (denoted as \(H_0\)) represents a statement of no effect or no difference. Essentially, it’s the default assumption that there is no change or no difference in the population from a specific value. In the case of the exercise, the null hypothesis posits that the proportion of families eating dinner together seven nights a week has not decreased and remains at 38%. Formally, we can write it as: \ \( H_0: p = 0.38 \).The null hypothesis serves as a starting point for statistical analysis, and it is tested against the alternative hypothesis to determine if the observed data provides enough evidence to reject it.Understanding the null hypothesis helps in setting the stage for hypothesis testing and ensures we are clear on the baseline we are comparing against.

Contrary to the null hypothesis, the alternative hypothesis (denoted as \(H_a\)) represents a statement that indicates the presence of an effect or difference. It is essentially what the researcher aims to prove. In our example, the alternative hypothesis suggests that the proportion of families eating dinner together seven nights a week has decreased from 38%. We express this as: \ \( H_a: p < 0.38 \).The alternative hypothesis is critical because it provides a direction or claim that we are trying to find evidence for. When the null hypothesis is rejected based on the data, it supports the acceptance of the alternative hypothesis. This shift is fundamental in hypothesis testing, as it moves us to acknowledging a new understanding based on statistical evidence.

The sample proportion, represented as \( \hat{p} \), is a statistic that estimates the proportion of a characteristic in a population, based on a sample from that population. In the exercise, the sample proportion is calculated from the data as follows:

• The number of families who eat dinner together 7 nights a week is 337.
• The total number of surveyed families is 1122.

Using the formula \( \hat{p} = \frac{337}{1122} \), we get \( \hat{p} = 0.3005 \). This means that approximately 30.05% of the surveyed families report eating dinner together every night.Sample proportion is crucial in hypothesis testing as it provides an estimate to compare against the assumed population proportion in the null hypothesis.

The test statistic in hypothesis testing is a standardized value that is calculated from the sample data. It helps in determining how far the sample statistic is from the population parameter specified in the null hypothesis. For proportions, the test statistic is given by the formula: \ \[ z = \frac{ \hat{p} - p }{ \sqrt{ \frac{p(1-p)}{n} } } \] Where:

• \( \hat{p} \) is the sample proportion (0.3005 in this example).
• \( p \) is the population proportion under null hypothesis (0.38).
• \( n \) is the sample size (1122).

Using the values from our exercise, we get: \ \[ z = \frac{0.3005 - 0.38}{ \sqrt{ \frac{0.38 \times 0.62}{1122} } } \approx -5.06 \] This negative z-score indicates how many standard deviations the sample proportion is below the population proportion. A large magnitude suggests a significant deviation, leading us to question the null hypothesis.

The significance level, denoted by \( \alpha \), is a threshold set by the researcher to determine the probability of rejecting the null hypothesis when it is actually true. Commonly set at 0.05, it represents a 5% risk of concluding that a difference exists when there is none.In our example, the significance level is \( \alpha = 0.05 \). To conclude if the null hypothesis can be rejected, we compare the p-value obtained from the test statistic to this threshold. If the p-value is less than \( \alpha \), we reject the null hypothesis.In the given exercise, the p-value (close to 0 due to a z-score of -5.06) is far below 0.05, leading us to reject the null hypothesis. Therefore, we find significant evidence to suggest that the proportion of families eating dinner together 7 nights a week has indeed decreased.