# Hypothesis Test Elections

Can you please tell me if this is right?

Suppose that, from the previous elections in a state, in order for a candidate to the governor to be elected, it is necessary for him to win at least 80% of the votes in the northern section of the state. The governor on duty is interested in assessing his chances of returning to the office and plans to conduct a survey of 2,000 registered voters in the northern section of the state. Apply the procedure to test hypotheses and evaluate the governor's chances of being re-elected, with a significance level of 0.05, if the poll revealed that 1,550 of the 2,000 possible voters in the northern part of the state planned to vote for the governor in turn. Is the sample results close enough to expected to conclude that the difference is due to the sampling error?

$$H_0: p\geq0.80$$ $$H_a: p<0.80$$

$$\alpha=0.05$$

The critical value: $$z=1.645$$

$$\hat{p}=\frac{1,550}{2,000}=0.775$$

$$p=0.80$$

$$q=1-p=1-0.8=0.20$$

Z-statistic:

$$z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}$$

$$z=\frac{0.775-0.80}{\sqrt{\frac{0.80(0.20)}{2,000}}}=-2.80$$

$$z=-2.80$$ falls in the reject region.

So the null hypothesis is rejected in the level of confidence of 0.05.

80% - 77.5% = 2.5% results significative statistically.

In other words, the evidence does not support the claim that the governor on duty returns to his mansion for another four years.

Is this analysis right?

• Remarks: this is the 95% confidence level, not the 5% confidence level. We also usually assume the null hypothesis is of the form "equal to," but it doesn't really matter much. – Sean Roberson Oct 15 '19 at 3:13
• I originally took the null with the equal. But I was confused with the at least that for me is greater or equal. So I wasn't sure where to put it, if in null or alternate. If I put the null with the equal sign the alternate hypothesis still will be less? – gi2302 Oct 15 '19 at 4:56

The calculations are correct but the phrasing of the conclusion is a bit awkward. Here, the polling data does furnish sufficient evidence that the governor's reelection is at risk, at a $$95\%$$ confidence level. That is to say, the sampling proportion of $$\hat p = 0.775$$ is sufficiently far below $$0.8$$ for the sample size of $$n = 2000$$ that the probability of having observed a result as low as this by random chance when his actual support is at least $$80\%$$, is less than $$0.05$$.