Z-Score and Proportion Activity 1. The researcher found that stress ratings varied per person and wanted to reach out to those most stressed and least stressed to find out why. However, all the data are based on different scales (for example two scales were "out of 100 points" while the other was "out of 200 points"). The higher the scores on each scale indicate more stress, but the only way to compare all data would be to calculate Z scores to see how many standard deviations (SD) points each student's score (X) is from the mean (M) of their group. Calculate the three random students' Z scores and report who is "most stressed" and "least stressed". To find the Z score that corresponds to a raw score, use the following formula. Z -X - M X is "score", M is "mean" (average), SD is Standard Deviation (sometimes seen as just S) SD Student A X = 10 M = 60 SD = 10 Z= Student B X = 75 M = 50 SD = 5 Z = Student C X = 110 M = 100 SD = 20 Z = The student who is MOST stressed is Student The student who is LEAST stressed is Student 2. What if it's the Z score we know and we want to find out what the raw score was? To find the raw score that corresponds to a Z score, use the following formula. X = (Z)(SD) + M Z is "Z score", SD is Standard Deviation (sometimes seen as just S), M is "mean" (average) Student D Z = 2.0 SD = 10 M = 150 X= Student E Z= -1.5 SD = 5 M = 100 X = Student F Z = 1.0 SD = 20 M = 70 X = 3. To put the scores in perspective based on the bell curve, determine the appropriate proportions of scores either above or below the given Z score. The proportion of scores below a Z-score of +2.60 = The proportion of scores below a Z-score of -0.36 = The proportion of scores above a Z-score of +0.55 = The proportion of scores above a Z-score of -0.75 = The Z score that corresponds to the bottom 6% of the scores = The Z score that corresponds to the top 14% of the scores = Example of Body vs Tail (big area" = Body, "Tiny area" = Tail Tips on Reading the Z table: Positive Z looking ABOVE the value, look in TAIL column Positive Z looking BELOW the value, look in BODY column Negative Z looking ABOVE the value, look in BODY column BODY TAIL Negative Z looking BELOW the value, look in TAIL column1Variance and Standard Deviation Activity The researcher wanted to determine how many dain interactions with a pet it needed to reduce stress, though needed to nd out how manyr times were typical in the population. However, this information needs to be estimated based on a sample of pet owners from the population. Eight individuals were sampled and asked how many times per day they played with their pet. Consider the following data and complete the following calculations [showing all of your work including the formula you are using]: 4 5 2 S T IIS 1 9 i. What is the range Xmas. Xmin = 2. What is the sum of squares {SS}? You can calculate the SS so that you can gure out the variance (32} and standard deviation {3} for these data as a population and as a sample. For now. it has been done for you. Population Formula for SS \"t- 2 E- 45.25 SS = Z (X ,u} 5-5.2: [-35.55- Sam le Formula for SS li-mli' p 7-5.25 ss _ 20f _ M): e-eaatl _ _ 1-5.25 _ 9-5.2: 3. What is the variance if this was a \"population\"? 4. What is the standard deviation if this was a "population\"? 5. What is the variance if this was a \"sample"? 6. What is the standard deviation if this was a \"sample"? 7. Recap question: Calculate andlor explain the mean, median and mode of the data above. 4 5 2 E T 6 1 9 put in numerical order -) Mean = Median = Mode = hint... there would need to he a number that repeats itself to have a mode. So in this case what would you say? Reminder Notes.- The \"standard deviation" is the "square root of the variance" The \"variance\" is the \"standard deviation squared"