A newspaper claims that 52% of voters will vote for a particular candidate in an upcoming election. A random survey of 1000 voters shows that 48% will vote for that candidate, with a standard deviation of 3%.
What is your question?
what is the answer?
To determine whether the newspaper's claim of 52% voter support for a particular candidate is accurate, we can conduct a hypothesis test using the information provided by the random survey.
Here's how you can perform the hypothesis test:
1. State the hypotheses:
- Null hypothesis (H0): The true proportion of voters supporting the candidate is 52%.
- Alternative hypothesis (Ha): The true proportion of voters supporting the candidate is not equal to 52%.
2. Determine the significance level (α) for the test. This value indicates how confident we want to be in rejecting the null hypothesis. Let's assume α = 0.05, which is commonly used as a threshold in hypothesis testing.
3. Calculate the standard error of the proportion:
The standard error, denoted as SE, can be calculated using the formula: SE = √[(p * q) / n], where
- p is the sample proportion (48% or 0.48),
- q is the complementary proportion (1 - p),
- n is the sample size (1000).
SE = √[(0.48 * 0.52) / 1000] = √[(0.2496) / 1000] = √0.0002496 ≈ 0.0158
4. Calculate the z-score:
The z-score measures how many standard errors the observed proportion (sample proportion) is away from the hypothesized proportion (52%). It can be calculated using the formula: z = (p - P) / SE, where
- p is the sample proportion (48% or 0.48),
- P is the hypothesized proportion (52% or 0.52),
- SE is the standard error of the proportion (0.0158).
z = (0.48 - 0.52) / 0.0158 = -0.04 / 0.0158 ≈ -2.53
5. Determine the critical value(s) and the rejection region:
The critical value(s) depend on the significance level (α) and the type of test (one-tailed or two-tailed).
Since our alternative hypothesis is two-tailed (not equal to 52%), we need to find the critical values at both ends of the distribution. At α = 0.05, the critical values can be obtained from a standard normal distribution table or using a calculator. For a two-tailed test, the critical values are ±1.96.
6. Compare the z-score to the critical value(s):
If the calculated z-score falls within the rejection region (outside the critical values), we reject the null hypothesis. Otherwise, if the z-score is within the non-rejection region, we fail to reject the null hypothesis.
In this case, the calculated z-score of -2.53 is less than -1.96 (the critical value for a 95% confidence level in a two-tailed test). Therefore, the z-score falls within the rejection region.
7. Make a conclusion:
Since the calculated z-score falls within the rejection region, we reject the null hypothesis. This means we have evidence to suggest that the true proportion of voters supporting the candidate is not 52%.
It's important to note that hypothesis tests provide evidence, but they do not prove or disprove a claim with absolute certainty. The results of this test suggest that the newspaper's claim is not supported by the sample data, but further investigation and analysis may be required.