Need help. Please answer the following questions. Kindly show your solutions. Thank you.

The z - score The areas under the normal curve are given in terms of z - values or scores. Either the z -score locates X within a sample or within a population. The formula for calculating z is: Z = AH (z-score for population data) Z = X-X (z - score for sample data) Where: X = are given measurement H = population mean " = population standard deviation X = sample mean s = sample standard deviation What is the importance of the z-score? Raw scores may be composed of large values, but large values cannot be accommodated at the base line of the normal curve So, they have to be transformed into scores for convenience without sacrificing meanings associated with the raw scores Recall that in the previous chapter, the graph of random variables locates the X scores on the x- axis. In mathematics, these locations are called zeroes. We connect this concept t the normal curve concept and we call our standard deviations z (for zero) scores. For any population, the mean and the standard deviation are fixed. Thus, the z formula matches the z - values one - to - one with the X values (raw scores). That is, for every X value there corresponds a z - value and for each z - value there is exactly one X value. The z - values are matched with specific areas under the normal curve in a normal distribution table. Therefore, if we wish to find the percentage associated with X, we must find its matched z - value using the z - formula. The z - value leads to the area under the curve found in the normal curve table, which is a probability, and that probability gives the desired percentage for X. The following examples illustrate these concepts. 1. Reading Scores Given the mean, u = 50 and the standard deviation, o = 4 of a population of Reading Scores. Find the z - value that corresponds to a score X = 58. Steps Solution 1. Use the computing formula for finding z - Z = X- H scores of population data. 2. Check the given values. Since these are u = 50, 0 = 4, and X = 58. population values, the z - score locates X within a population. 3. Substitute the given values in the computing 8 Z = - - = 2 formula. Thus, the z - value that corresponds to the raw score 58 is 2 in a population distribution. This conversion from raw score to z - score is shown graphically in figure below. From the diagram, we see that a score X = 58 corresponds to z = 2. It is above the mean. So we can say that, with respect to the mean, the score of 58 is above average. Note that in figure above, because z = 0 is the center of the distribution, the negative z - values simply indicate that these values are found at the left of the center. 2. Score in PE Locate the z - value that corresponds to a PE score of 39 given that u = 45 and o = 6. Steps Solution 1. Use the corresponding formula for finding z - X - H Z = scores of the population data. 2. Check the given values. The z - score in 1=45, 6 = 6, and X = 39. question locates X in a population. 3. Substitute the given values in the computing 39 - 45 formula. -6 Z = - 4. Compute the z - value. Thus, the z value that corresponds to the raw score 39 is -1 in a population distribution.Steps Solution 1. Use the corresponding formula for nding 2- X p scores of the population data. 0' 2. Check the given values. The 2 - score In question locates X in a population. 3. Substitute the given values in the computing 39 4-5 formula. 2 _ 6 6 Z = ? = 1 4' Compute the z ' value. Thus, the 2 value that corresponds to the raw score 39 is -1 in a population distribution. With respect to the mean, the score 39 is below the population mean. We can also say that the score 39 is below average. 3. Scores in a Science Test Given X = 20.1: 25 and s = 4. Compute the corresponding 2- score. Steps Solution 1. Use the computing formula for finding 2 - scores X X of sample data. 2 = .s 2. Check the given values. The 2- score in question locates x in a sample. 3. Substitute the given values in the computing 20 26 formula. 2 _ {=26,s=4andx=20. 6 z== = 1.5 lerb 4. Compute the 2- value. The corresponding 2- score is -1.5 to the left of the mean. Explore your understanding: State whether the z - score locates the raw score x within a sample or within a population. 1.)l=82, o'= 15,,u=75 2.x:5u, s=s,=4u 3.)l=74, s= 10,K=60 Firm - Up your knowledge: From Explore, state whether each raw score lies below or above the mean. Deepen your understanding: Given K = 62 and s = 8. Find the z- score value that corresponds to each of the following scores up to two decimal places: 1. X = 70 2. X = 78 3. )l = 82 Transfer: The scores of students in the midyear examination for Mathematics are 37 and 22 and it has a mean (u) of 32 and a standard deviation (0') of 5. Find the z - score