Question #51184

(a) If x is the sample mean, prove that the expected value of x,E(x) equals the population mean (μ).
(b) Describe the process of testing hypothesis about population proportion of a given attribute.

Expert's answer

(a) The expected value of the sampling distribution of the sample mean equals the populations mean for all populations. Using advanced mathematics, in a thought experiment, the theoretical statistician often discovers a relationship between the expected value of a statistic and the model parameters. For example, it can be proven that the expected value of both the mean and the median is equal to μ. When the expected value of a statistic equals a population parameter, the statistic is called an unbiased estimator of that parameter. In this case, both the mean and the median would be an unbiased estimator of the parameter μ.

(b) The process of hypothesis testing involves setting up two competing hypotheses, the null hypothesis and the alternate hypothesis. One selects a random sample (or multiple samples when there are more comparison groups), computes summary statistics and then assesses the likelihood that the sample data support the research or alternative hypothesis. Similar to estimation, the process of hypothesis testing is based on probability theory and the Central Limit Theorem.

(b) The process of hypothesis testing involves setting up two competing hypotheses, the null hypothesis and the alternate hypothesis. One selects a random sample (or multiple samples when there are more comparison groups), computes summary statistics and then assesses the likelihood that the sample data support the research or alternative hypothesis. Similar to estimation, the process of hypothesis testing is based on probability theory and the Central Limit Theorem.

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