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The Z-score serves several purposes in statistics: Standardization: It allows for the comparison of scores from different distributions or datasets by standardizing the values. This is particularly useful when dealing with data from different sources or scales. Understanding the Position of a Value: A Z-score indicates how far and in what direction a value deviates from the mean. For example, a Z-score of 2 means the value is two standard deviations above the mean, while a Z-score of -1.5 means the value is one and a half standard deviations below the mean. Probability and Statistical Significance: In a normal distribution, Z-scores can be used to calculate the probability of a value occurring under the curve. They are also used in hypothesis testing to determine the statistical significance of a result. Identification of Outliers: Extreme Z-scores (typically those greater than 3 or less than -3) can indicate that a value is an outlier, or significantly different from the rest of the data. Biostatistical Problem: In a biostatistics study, researchers are investigating the average level of a certain protein in the blood of individuals with a specific health condition. The protein levels in this population are normally distributed. From previous studies, it is known that the mean protein level in this population is 150 units with a standard deviation of 20 units. A patient named Alex participates in the study, and his blood test shows a protein level of 175 units. Calculate Alex's Z score for his protein level and interpret the result.
