SOLUTION: A researcher claims that the Republican Party is going to win in the next Senate elections especially in Florida State. A statistical data stated that 23% of the voters opted for t

Question 1169056: A researcher claims that the Republican Party is going to win in the next Senate elections especially in Florida State. A statistical data stated that 23% of the voters opted for the Republican Party in the last election. In order to test the claim, an investigator surveyed 80 people and found 22 of them voted for Republican Party in the last election held. Is there enough statistical evidence at α = 0.05 to support the claim of the researcher?

Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
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.
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