A politician claims that she will receive 59% of the vote in an upcoming election. The results of a properly designed random sample of 100 voters showed that 54 of those sampled will vote for her. Is it likely that her assertion is correct at the 0.05 level of significance? What is Z? What is p-Value?
To determine whether the politician's claim that she will receive 59% of the vote is likely to be correct, we need to conduct a hypothesis test and calculate the z-score and p-value.
Let's set up the hypotheses for our test:
- Null Hypothesis (H0): The politician's claim is correct, and she will receive 59% of the vote.
- Alternative Hypothesis (Ha): The politician's claim is not correct, and she will not receive 59% of the vote.
Now, let's calculate the z-score, which measures how many standard deviations the sample proportion is away from the claimed proportion.
The formula for calculating the z-score is:
z = (p̂ - p) / sqrt(p * (1 - p) / n)
Where:
p̂ is the sample proportion (54/100 = 0.54),
p is the claimed proportion (0.59),
n is the sample size (100).
Substituting these values into the formula:
z = (0.54 - 0.59) / sqrt(0.59 * (1 - 0.59) / 100) ≈ -1.02
The z-score in this case is approximately -1.02.
Next, we need to calculate the p-value, which represents the probability of obtaining a sample proportion as extreme as the one observed, assuming the null hypothesis is true.
To find the p-value, we need to calculate the area under the normal distribution curve beyond the absolute value of the z-score we obtained.
The p-value can be found using a standard normal (Z) table or using statistical software. The p-value for a two-tailed test is the probability of obtaining a z-score as extreme as -1.02 or more extreme in either tail.
Using a Z-table or statistical software, we find the p-value to be approximately 0.307.
Now, let's interpret the results based on the significance level of 0.05:
Since the p-value (0.307) is greater than the significance level (0.05), we fail to reject the null hypothesis. There is not enough evidence to conclude that the politician's claim is incorrect. However, it is important to note that this does not prove the claim is true; it simply means there is not enough evidence to disprove it at the 0.05 level of significance.
In summary:
- Z-score: -1.02
- p-value: 0.307