A student of the author surveyed her friends and found that among 20 males, 4 smoke and among 30 female friends, 6 smoke. Give two reasons why these results should not be used for a hypothesis test of the claim that the proportions of male smokers and female smokers are equal.
„h Given a simple random sample of men and a simple random sample of women, we want to use a 0.05 significance level to test the claim that the percentage of men who smoke is equal to the percentage of women who smoke. One approach is to use the P-value method of hypothesis testing; a second approach is to use the traditional method of hypothesis testing; and a third approach is to base the conclusion on the 95% confidence interval estimate of p1¡Xp2. Will all three approaches always result in the same conclusion? Explain.
I need help to understand formula
The normal approximation may be inaccurate for small samples.Here bith the samples are real;y small sizes, that is why you should not risk the hypothesis testing.
Also there is no information of population size and distribution, which may stop us from finding if we need to use finite population correction, and assuming these distribution to be normal would be a far stretched assumption without any proof.
There are two main reasons why the results from the survey should not be used for a hypothesis test of the claim that the proportions of male smokers and female smokers are equal:
1. Sample Size: The proportions of smokers among the surveyed males and females are based on relatively small sample sizes of 20 and 30, respectively. Small sample sizes may not accurately represent the true proportions of smokers in the population. To obtain more reliable results, a larger sample size is generally preferred.
2. Sampling Method: The survey was conducted by a student who surveyed her own friends. This sampling method introduces potential biases and may not provide a representative sample of the entire population. Ideally, a random sampling method should be used to select participants to ensure that the sample is representative of the population as a whole.
Now, regarding the three approaches - P-value method, traditional method, and 95% confidence interval estimate of p1-p2 - it is important to note that each approach has its own assumptions and interpretations. The conclusion may not always be the same for all three approaches. Here's a brief explanation of each approach:
1. P-value method: In this approach, a p-value is calculated based on the observed data, and if the p-value is less than the chosen significance level (in this case, 0.05), then the null hypothesis (that the proportions of male smokers and female smokers are equal) is rejected. The conclusion is based on the strength of the evidence against the null hypothesis.
2. Traditional method: This approach involves calculating a test statistic (such as the z-statistic or chi-square statistic) based on the observed data and comparing it to the critical value from the corresponding distribution at the chosen significance level. The conclusion is based on whether the test statistic falls in the rejection region or not.
3. 95% confidence interval estimate of p1-p2: This approach involves calculating a confidence interval for the difference in proportions (p1-p2) using the observed sample proportions. If the confidence interval does not contain zero, then it indicates evidence against the null hypothesis. The conclusion is based on whether the interval includes or excludes the value of zero.
Since each approach has different calculations and interpretations, it is possible that they may lead to different conclusions. However, if the sample sizes were larger and the sampling method was more representative, the conclusions from these approaches would likely converge towards a similar result.