Introduction to Statistics course :

here is the question

1-The Standard Normal Distribution

note* The solution from this pictures

6.1 | The Standard Normal Distribution The standard normal distribution is a normal distribution of standardized values called z-scores. A z-score is measured in units of the standard deviation. For example, if the mean of a normal distribution is five and the standard deviation is two, the value 11 is three standard deviations above (or to the right of) the mean. The calculation is as follows: x = u+(z)(o) = 5 + (3)(2) =11 This OpenStax book is available for free at http://cnx.org/content/col11562/1.18 Chapter 6 | The Normal Distribution 367 The z-score is three. The mean for the standard normal distribution is zero, and the standard deviation is one. The transformation z = produces the distribution Z ~ N(0, 1). The value x in the given equation comes from a normal distribution with mean / and standard deviation o. Z-Scores If X is a normally distributed random variable and X ~ N(p, o), then the z-score is: X - H The z-score tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, p. Values of x that are larger than the mean have positive z-scores, and values of x that are smaller than the mean have negative z-scores. If x equals the mean, then x has a z-score of zero.Example o.1 Suppose X ~ N(5, 6). This says that X is a normally distributed random variable with mean p = 5 and standard deviation o = 6. Suppose x = 17. Then: 2-*-M _17-5 -2 6 This means that x = 17 is two standard deviations (20) above or to the right of the mean p = 5. Notice that: 5 + (2)(6) = 17 (The pattern is u + zo = x) Now suppose x = 1. Then: z = = 1-> = -0.67 (rounded to two decimal places) This means that x = 1 is 0.67 standard deviations (-0.670) below or to the left of the mean , = 5. Notice that: 5 + (-0.67)(6) is approximately equal to one (This has the pattern p + (-0.67) = 1) Summarizing, when z is positive, x is above or to the right of u and when z is negative, x is to the left of or below p. Or, when z is positive, x is greater than , and when z is negative x is less than . Try It & 6.1 What is the z-score of x, when x = 1 and X ~ N(12,3)? Example 6.2 Some doctors believe that a person can lose five pounds, on the average, in a month by reducing his or her fat intake and by exercising consistently. Suppose weight loss has a normal distribution. Let X = the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of two pounds. X ~ N(5, 2). Fill in the blanks. a. Suppose a person lost ten pounds in a month. The z-score when x = 10 pounds is z = 2.5 (verify). This z-score tells you that x = 10 is standard deviations to the (right or left) of the mean (What is the mean?). Solution 6.2 a. This z-score tells you that x = 10 is 2.5 standard deviations to the right of the mean five. b. Suppose a person gained three pounds (a negative weight loss). Then z = This z-score tells youChapter 6 | The Normal Distribution that x = -3 is standard deviations to the (right or left) of the mean. Solution 6.2 b. z =-4. This z-score tells you that x = -3 is four standard deviations to the left of the mean. c. Suppose the random variables X and Y have the following normal distributions: X ~ N(5, 6) and Y ~ N(2, 1). If x = 17, then z = 2. (This was previously shown.) If y = 4, what is z? Solution 6.2 C. Z = = y - H _ 4 -2 = 2 where u = 2 and o = 1. The z-score for y = 4 is z = 2. This means that four is z = 2 standard deviations to the right of the mean. Therefore, x = 17 and y = 4 are both two (of their own) standard deviations to the right of their respective means. The z-score allows us to compare data that are scaled differently. To understand the concept, suppose X ~ N(5, 6) represents weight gains for one group of people who are trying to gain weight in a six week period and Y ~ N(2, 1) measures the same weight gain for a second group of people. A negative weight gain would be a weight loss. Since x = 17 and y = 4 are each two standard deviations to the right of their means, they represent the same, standardized weight gain relative to their means