Question 1: Please define a Z-Score? Question 2: What are the main differences b

Question

Question 1: Please define a Z-Score?
Question 2: What are the main differences between a Z-Socre and an X-Probability? What are their correlations?
Question 3: In a Bell-Shaped Distribution what is the main purpose of a Z-Score? Please provide an example to support your answers.
Question 4: Please explain degrees of freedom?
Question 5: Please provide an example of a probability equal to one, and a probability equal to zero. Why a probability can't be negartive? Explain in detail to support your answers.

Explanation / Answer

1.) z-score or the standard score is the number of standard deviations from the mean a data point is. It's a measure of how many standard deviations below or above the population mean a raw score is. andard score) indicates how many standard deviations an element is from the mean. A z-score can be calculated from the following formula. z = (X - ) / where z is the z-score, X is the value of the element, is the population mean, and is the standard deviation.

2.) X-porbability can be defined as the element in the data set. While z-score the measure of number of standard deviations the element is from the mean of the data set. z = (X - ) / where z is the z-score, X is the value of the element, is the population mean, and is the standard deviation.

3.) The normal distribution is a symmetrical, bell-shaped distribution in which the mean, median and mode are all equal. It is a central component of inferential statistics. The standard normal distribution is a normal distribution represented in z scores. It always has a mean of zero and a standard deviation of one. We can use the standard normal table to calculate the area under the curve between any two points. Think of z scores as just another unit of measurement. If, for example, we were measuring time, we could express time in terms of seconds, minutes, hours or days. Similarly we could measure distance in terms of inches, feet, yards or miles.  

4.) The degrees of freedom in a statistical calculation represent how many values involved in a calculation have the freedom to vary. The degrees of freedom can be calculated to help ensure the statistical validity of chi-square tests, t-tests and even the more advanced f-tests. These tests are commonly used to compare observed data with data that would be expected to be obtained according to a specific hypothesis.

For example, let's suppose a drug trial is conducted on a group of patients and it is hypothesized that the patients receiving the drug would show increased heart rates compared to those that did not receive the drug. The results of the test could then be analyzed to determine whether the difference in heart rates is considered significant, and degrees of freedom are part of the calculations.

Because degrees of freedom calculations identify how many values in the final calculation are allowed to vary, they can contribute to the validity of an outcome. These calculations are dependent upon the sample size, or observations, and the parameters to be estimated, but generally, in statistics, degrees of freedom equal the number of observations minus the number of parameters. This means there are more degrees of freedom with a larger sample size.

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