Given that there are total friends and said they are voting for candidate A. So,
The sample proportion of votes to candidate A is given by;
1. Sara concluded that candidate A will get more than 50% votes.
Sara’s conclusion may not be true as sample size is low also in her friend circle may be most of the people have same mind set. So, this sample became non-random hence conclusion became statistically wrong. This can also be tested the hypothesis approach.
The test statistics is given by
Where is standard normal distribution. Since this test is one tailed, testing the P-value is given by;
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Since the P-value is and comparing it with level of significance (95% confidence level) we fail to reject the null hypothesis as P-vale>level of significance.
Hence Sara conclusion is statistically not true at 0.05 level of confidence.
2. Let the sample size of friends that Sara need be n. To draw her conclusion with 95% confidence interval. We use the formula where ME is the margin error at 95% confidence interval.
To draw Sara’s conclusion at 95% confidence interval should contain proportions above 0.50, that is, the lower limit of 95% confidence level should be greater than 0.5 hence
Since margin of error is given by
Where defined as . Now as . Hence
So, Sara needs a friends sample of size .
3. Sara’s conclusion may be wrong if candidate A doesn’t get more than 50% votes. From a large number of voters, Sara asked only 15 people that they are going to vote of candidate A or candidate B. Therefore, its impractical to say that only 15 people represent the decision of large number of voters. We can say that Sara needs more than 15 people for the validity of her conclusion.
References
Fleiss, J. L., Levin, B., & Paik, M. C. (2013). Statistical methods for rates and proportions . john wiley & sons.
Gupta, S. K. (2012). The relevance of confidence interval and P-value in inferential statistics. Indian journal of pharmacology , 44 (1), 143.
Kalbfleisch, J. G. (2012). Probability and statistical inference . Springer Science & Business Media.
