Question

7. z-scores and standardized scores Is a z-score a standardized score? No Yes Consider the following distribution of scores with a mean of 50 and a standard deviation of 10. For the letters A, B, C, and D in the boxes beneath the ine labeled "z" give the z-scores corresponding to each position in the distribution. One z-score is already filled in (-1) Suppose you also want to standardize these scores to a "k" scale where the mean of k is 100 and the standard deviation is 20·For the letters E, F, G, and H in the boxes beneath the line labeled "k" give the k-scores corresponding to each position in the distribution. One k-score is already filled in (120)

Answer 1

A z-score is indeed a standardized score because a standardized score shows the number of standard deviation the data values lie above or below the mean. And, the z-score is the ratio of the difference between a score and mean to the standard deviation. This shows that the Z-score is a standardized score. The distribution of scores is having mean 50 and a standard deviation of 100. Thus, u=5 o=10 The Z-score is written as follows: x-u 2 x-50 10 For A, the value x = 30. Thus, the corresponding z-score is, x - u 30-50 10 =-2

For B, the value x = 50. Thus, the corresponding Z-score is, x-u 2=- 50 - 50 10 = 0 For C, the value x = 60. Thus, the corresponding z-score is, X-U 60-50 10 For D, the value x = 70. Thus, the corresponding Z-score is, :=x-4 70 – 50 10

Now the z- scores obtained above are to standardize to a “k” scale with a mean 100 and a standard deviation 20. Thus, u = 100 O = 20 The value x is obtained from the z-score as follows: X -u > x = u + 20 → x =100+ 20- For the value of Z-score A=-2, the value of x is, X- u Z =- => x = 100+20(-2) => x = 60 For the value of z-score -1, the value of x is, :-*-u x=100 >> x = 80

For the value of z-score B=0, the value of x is, ==x-u x=100+20(0) x=100 For the value of z-score C =1, the value of x is, X-u 2 =- →x=100+ 20(1) => x=120 For the value of z-score D= 2, the value of x is, x-u =x=100+20(2) = x=140

Thus, the final graph showing all obtained values is as follows: 30 40 50 60 70 TE C=1 D=2 A=-2 60 -1 80 B=0 100 120 140 . Hence, the above-obtained shows all the obtained values.