Question 1169056
To determine if there is enough statistical evidence to support the researcher's claim, we need to perform a hypothesis test for a population proportion.

**1. Define the hypotheses:**

* **Null Hypothesis ($H_0$):** The proportion of voters who opted for the Republican Party in the last election in Florida is equal to or less than 23% (the historical data).
    $H_0: p \le 0.23$
* **Alternative Hypothesis ($H_a$):** The proportion of voters who opted for the Republican Party in the last election in the surveyed group is greater than 23%. This would support the researcher's claim of increased Republican support.
    $H_a: p > 0.23$

**2. Set the significance level:**

The significance level is given as $\alpha = 0.05$.

**3. Collect and summarize the sample data:**

* Sample size ($n$) = 80
* Number of voters who opted for the Republican Party in the sample ($x$) = 22
* Sample proportion ($\hat{p}$) = $x/n = 22/80 = 0.275$

**4. Calculate the test statistic:**

Since the sample size is large enough ($n > 30$ and $np_0 \ge 10$, $n(1-p_0) \ge 10$), we can use the z-test for proportions. The test statistic is calculated as:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$

where:
* $\hat{p}$ is the sample proportion (0.275)
* $p_0$ is the hypothesized population proportion under the null hypothesis (0.23)
* $n$ is the sample size (80)

Plugging in the values:

$z = \frac{0.275 - 0.23}{\sqrt{\frac{0.23(1 - 0.23)}{80}}}$
$z = \frac{0.045}{\sqrt{\frac{0.23 \times 0.77}{80}}}$
$z = \frac{0.045}{\sqrt{\frac{0.1771}{80}}}$
$z = \frac{0.045}{\sqrt{0.00221375}}$
$z = \frac{0.045}{0.04705}$
$z \approx 0.956$

**5. Determine the critical value:**

For a right-tailed test with a significance level of $\alpha = 0.05$, we need to find the z-score that corresponds to an area of 0.05 in the upper tail of the standard normal distribution. Using a z-table or a statistical calculator, the critical value $z_{\alpha} = z_{0.05} \approx 1.645$.

**6. Make a decision:**

We compare the calculated test statistic ($z \approx 0.956$) with the critical value ($z_{0.05} \approx 1.645$).

Since the test statistic ($0.956$) is less than the critical value ($1.645$), we fail to reject the null hypothesis.

**7. Draw a conclusion:**

At the $\alpha = 0.05$ level of significance, there is **not enough statistical evidence** to support the researcher's claim that the proportion of voters opting for the Republican Party has increased significantly in Florida since the last election. The observed sample proportion (27.5%) is higher than the historical proportion (23%), but the difference is not statistically significant enough to reject the possibility that it occurred due to random sampling variation.

It's important to note that this analysis is based on a single survey and only looks at the proportion of voters who *voted* Republican in the *last* election. It does not directly predict the outcome of the *next* Senate elections. The researcher's claim about the *next* election involves future events and potentially other factors not captured by this historical voting data.