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Hypothesis testing is a core part of inferential statistics, where we use sample data to make inferences about the larger population. In this problem, hypothesis testing is used to determine whether the sample proportion of students supporting a crackdown on underage drinking is significantly lower than the national proportion. Here's a breakdown of the process:

• **Null Hypothesis ( H_0 ):** This is the default assumption that there is no difference between the sample proportion and the population proportion. Here, the null hypothesis states that the proportion of support at the local college is equal to the national level, or 67%.
• **Alternative Hypothesis ( H_a ):** This is what you suspect might be true instead of the null hypothesis. In this scenario, the alternative hypothesis would suggest that the proportion of support at the local college is less than 67%.

Hypothesis testing provides a method to decide whether the data supports rejecting the null hypothesis in favor of the alternative hypothesis. This process plays a vital role when making informed decisions based on sample data.

The sample proportion is a key statistic when assessing the likelihood of a certain outcome in a survey or experiment. It represents the fraction of the sample showing the characteristic of interest.For our exercise:- The sample included 100 students, and out of these, 62 expressed support for the crackdown.- Thus, the sample proportion (\hat{p}) is calculated as the number of successes in the sample divided by the sample size, which gives us \( \hat{p} = \frac{62}{100} = 0.62 \).Understanding the sample proportion is crucial, as it forms the basis for further statistical calculations like the standard error and Z-score, which ultimately contribute to hypothesis testing.

The standard error (SE) is a measure that describes how much variability we might expect in the sample proportion compared to the true population proportion. It gives us an idea of how close our sample proportion is likely to be to the population proportion when sampling randomly.The standard error is calculated for a proportion using the formula:\[ \text{SE} = \sqrt{\frac{p(1-p)}{n}} \]where:

• \(p\) is the assumed population proportion (67% in this case).
• \(n\) is the sample size (100 students).

Using these values, the standard error is:\[ \text{SE} = \sqrt{\frac{0.67 \times 0.33}{100}} \approx 0.048 \]A smaller SE indicates that the sample proportion is a more accurate estimate of the population proportion.

The Z-score tells us how many standard deviations a data point is from the mean, providing a way to compare results from a typical standard distribution. It's essential for hypothesis testing to assess the rarity or extremity of the observed sample proportion.For our specific case:- **Formula:**\[ Z = \frac{\hat{p} - p}{\text{SE}} \]- **Substituting Values:** - Observed sample proportion \( \hat{p} = 0.62 \) - Population proportion \( p = 0.67 \) - Standard error \( \text{SE} = 0.048 \)- **Calculating Z-score:** \[ Z = \frac{0.62 - 0.67}{0.048} \approx -1.042 \]This Z-score can be looked up in a standard normal distribution table to find the probability of observing a value as extreme as the sample proportion. In this example, it indicates about a 14.9% chance that the sample proportion could be as extreme as 0.62 just by random chance, when the true proportion is indeed 67%. This further elucidates why we can't readily conclude there's less support locally without potentially large sampling errors.