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Case Study III: Inferential Statistics
This part of the case study will explore the application of inferential statistics to the Zagat Survey sample data. In Part II, claims were made that attempted to extrapolate the sample data to the population, but they were statistically invalid. So here in Part III, I will properly project the sample data into the population and compare it to the previous methods. Earlier in the study, the univariate estimate used was simply the mean value of the sample taken for the variable Cost. But this does not account for the presence of variation in a sample that could affect the mean. We can use this value within a standard normal distribution of sample means to try and narrow down the true population mean within a confidence interval of probability. This is preferable because it takes into account the standard error that is intrinsic to all iterated samples of a population and uses that to capture the population mean (μ) within a range. It is not plausible to calculate the probability that a continuous random variable will assume a specific value in a continuous probability distribution. To establish the boundaries of the confidence interval, we must first select the level of confidence (L). In this case, I will set L=0.98, its complement being a level of significance (α) of α=0.02. Since we only have the sample mean and standard deviation, we must use the t-table and find the critical value when α/2=0.01 (simple differentiation = two-tailed test). Appendix A establishes the variables and shows the calculation of the interval boundaries. This allows me to state with 98% confidence that the population mean Cost will fall between $39.14 and $44.76.
The management should not be concerned with the number of restaurants included in the sample. A larger sample would only lower the standard error and the t-value for the level of

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