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Let's assume that 40% of the nation is registered republican.  Does the Tahoe environment reflect the national proportion?  Test the hypothesis that Tahoe residents differ from the rest of the nation in their affiliation, if of 200 locals surveyed, 75 are registered republican. The 95% confidence interval is (0.30791, 0.44209). What should be our conclusion

Question

Let's assume that 40% of the nation is registered republican.  Does the Tahoe environment reflect the national proportion?  Test the hypothesis that Tahoe residents differ from the rest of the nation in their affiliation, if of 200 locals surveyed, 75 are registered republican. The 95% confidence interval is (0.30791, 0.44209). What should be our conclusion

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Solution

To answer this question, we need to perform a hypothesis test for a proportion. Here are the steps:

  1. State the hypotheses. The null hypothesis is that the proportion of Tahoe residents who are registered Republicans is the same as the national proportion, i.e., 0.40. The alternative hypothesis is that the proportion is different from 0.40.

  2. Calculate the test statistic. In this case, the test statistic is the difference between the sample proportion (75/200 = 0.375) and the hypothesized proportion (0.40), divided by the standard error of the sample proportion. The standard error is the square root of [(0.40 * (1 - 0.40)) / 200] = 0.02449. The test statistic is (0.375 - 0.40) / 0.02449 = -1.02.

  3. Determine the p-value. The p-value is the probability of observing a test statistic as extreme as -1.02 under the null hypothesis. Using a standard normal distribution (because we have a large sample size), we find that the p-value is 0.308.

  4. Compare the p-value to the significance level. The significance level is 0.05 (because we want 95% confidence). Because the p-value is greater than the significance level, we do not reject the null hypothesis.

  5. Interpret the result. There is not enough evidence at the 95% confidence level to conclude that the proportion of Tahoe residents who are registered Republicans is different from the national proportion.

However, the provided 95% confidence interval (0.30791, 0.44209) does include the national proportion of 0.40. This means that at a 95% confidence level, we can't reject the hypothesis that the proportion of Republicans in Tahoe is the same as the national proportion. So, both the hypothesis test and the confidence interval lead to the same conclusion.

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