91Ó°ÊÓ

In hypothesis testing, the null hypothesis, denoted as \( H_0 \), is the assumption that there is no effect or no difference, and it serves as the starting point for statistical testing. In simpler terms, it's like saying "nothing has changed."
For our specific exercise, the null hypothesis is that the proportion of married Americans is the same as it was in 2014, which is \( 62\% \).
To put it mathematically:

• The null hypothesis \( H_0: p = 0.62 \) means we assume that the current marriage rate is still \( 62\% \).

The opposing idea, called the alternative hypothesis, is what you propose if the null hypothesis is not supported by the data. Here, the alternative hypothesis suggests a decrease, stating:

• \( H_1: p < 0.62 \), meaning the marriage rate is less than \( 62\% \).

The significance level is a critical concept in hypothesis testing. It essentially determines how strict we are with rejecting the null hypothesis.
Typically, a significance level is denoted by \( \alpha \) and is stated as a percentage. In our example, the significance level is \( 1\% \) or \( 0.01 \) when expressed as a decimal.
This means there is only a \( 1\% \) risk of concluding that there is a difference when there is actually none. Essentially, it's the threshold we use to decide if our test results are due to chance or if they are statistically significant.
The significance level is crucial because it influences the determination of the critical value, which in turn affects whether we reject the null hypothesis.

A Z-score is a statistical measure that helps us understand where our data point lies in relation to the mean of a dataset. It's often used to test hypotheses when dealing with proportions or means.
In this exercise, we calculated the Z-score to test whether the sample proportion of married individuals (\( 58\% \)) differs from the population proportion (\( 62\% \)). The formula used is: \[Z = \frac{{\hat{p} - p}}{\sqrt{\frac{{p(1 - p)}}{n}}}\]

• \( \hat{p} = 0.58 \) is the sample proportion.
• \( p = 0.62 \) is the null hypothesis proportion.
• \( n = 2000 \) is the sample size.
• The calculated Z-score, \( -5.47 \), indicates how many standard deviations our sample mean is below the null hypothesis mean.

A negative Z-score here shows that the sample proportion is less than 62%, and given the critical Z value threshold for \( 1\% \) significance which is about \(-2.33\), our Z-score of \(-5.47\) is quite decisive.

The p-value is a crucial part of hypothesis testing, used to quantify the strength of the evidence against the null hypothesis. It's essentially the probability that the observed data would occur if the null hypothesis were true.
In this context, a very small p-value indicates strong evidence against the null hypothesis. Computed for this exercise using the calculated Z-score of \(-5.47\), the p-value is practically zero.
This means there's an extremely low probability that the observed proportion of married individuals (\(58\%\)) would occur if the true proportion were still \(62\%\).
The rule is simple:

• If the p-value is less than the significance level (and here it is \(0\) which is far less than our \(0.01\) significance), we reject the null hypothesis.

In conclusion, both the critical value approach and the p-value approach lead to rejecting the null in favor of the alternative, affirming that there's statistically significant evidence suggesting a lower marriage rate than \( 62\% \).