Question

Participants	Daily Calories	Z-Score ( Calories )	Life satisfaction	Z-Score ( Satisfaction )
1	2000	-0.082152426	10	1.472538182
2	3000	1.155082909	8	0.825268652
2	1500	-0.700770094	3	-0.792905175
4	1800	-0.329599493	10	1.472538182
5	3200	1.402529976	5	-0.145635644
6	1600	-0.57704656	4	-0.46927041
7	3100	1.278806442	1	-1.440174706
8	2500	0.536465241	6	0.177999121
9	2500	0.536465241	10	1.472538182
10	1900	-0.20587596	3	-0.792905175
11	1500	-0.700770094	4	-0.46927041
12	2900	1.031359375	7	0.501633886
13	3900	2.26859471	2	-1.11653994
14	2500	0.536465241	5	-0.145635644
15	3400	1.649977042	6	0.177999121
16	4000	2.392318243	0	-1.763809471
17	4200	2.63976531	8	0.825268652
18	2050	-0.020290659	7	0.501633886
19	2700	0.783912308	9	1.148903417
20	3030	1.192199969	1	-1.440174706

For the number of daily calories consumed and life satisfaction report the z-scores for participants number 2,6 and 7 . Discuss where these participants scored in relation to the mean and explain how you know the information.

Calculate what raw score was equal to a z score of -1

Using the z table explain what percentage of the distribution scored below this score

Calculate z-scores for these participants 2,6,7 using the z-score formula

Rearrange the formula to figure out what raw scores corresponds with a Z-SORE OF -1

Answer 1

We know that z score = (x-mean)/standard deviation

that is x = mean z score * standard deviations

Thus if z score is positive , x is z score * standard deviations times more than mean

if z score is negative  , x is z score * standard deviations times less  than mean

In short , if x is more than mean z score is positive or else negative

Here x means number of daily calories / life satisfaction

For participant number 2 , z score for Calories is  1.155082909 . That is participant number 2 scored 1.16 times standard deviations more than mean .

For participant number 2 , z score for Life satisfaction is 0.825268652 . That is participant number 2 scored 0.83 times standard deviations more than mean .

For participant number 6 , z score for Calories is -0.577... . That is participant number 6 scored 0.58 times standard deviations less than mean .

For participant number 6 , z score for Life satisfaction is -0.469.... . That is participant number 6 scored 0.47 times standard deviations less than mean .

For participant number 7 , z score for Calories is 1.2788... . That is participant number 6 scored 1.28 times standard deviations more than mean .

For participant number 7 , z score for Life satisfaction is -1.44.... . That is participant number 7 scored 1.44 times standard deviations less than mean .

z score =-1

raw score =?

Let us find raw score of daily calories

for participant number 2

raw score , x = 3000 , z score = 1.155082909

  

for participant number 6

raw score , x = 1600 , z score = -0.57704656

Solving two equations , we get

s= 808.25

Thus for z score = -1

raw score = 2066.40 -808.25 = 1258.15

Let us find raw score of life satisfaction  

for participant number 2

raw score , y=8 , z score =0.8252686

  

for participant number 6

raw score , y=4 , z score = -0.46927041

Solving two equations , we get

s= 3.09

Thus for z score = -1

raw score = 5.45-3.09 = 2.36

From z table , we know that P( z < -1 ) = 0.1587

Thus 15.87 % scored below 1258.15 (daily calories)

15.87% scored below 2.36 ( life satisfaction)

To find z score for participants

Let us find mean and standard deviation from the given sample

mean

standard deviation

	x	(x-2664)^2		y	(y-5.45)^2
	2000	440896		10	20.7025
	3000	112896		8	6.5025
	1500	1354896		3	6.0025
	1800	746496		10	20.7025
	3200	287296		5	0.2025
	1600	1132096		4	2.1025
	3100	190096		1	19.8025
	2500	26896		6	0.3025
	2500	26896		10	20.7025
	1900	583696		3	6.0025
	1500	1354896		4	2.1025
	2900	55696		7	2.4025
	3900	1527696		2	11.9025
	2500	26896		5	0.2025
	3400	541696		6	0.3025
	4000	1784896		0	29.7025
	4200	2359296		8	6.5025
	2050	376996		7	2.4025
	2700	1296		9	12.6025
	3030	133956		1	19.8025
sum	53280	13065480		109	190.95
mean	2664			5.45
s^2		687656.8			10.05
s		829.2508			3.170173

Thus

Therefore

and

Participant	x	z score	y	z score
2	3000	0.405185	2	-1.08833
6	1600	-1.28309	4	-0.45741
7	3100	0.525776	1	-1.40379

z score = -1

raw score , x = -1 * 829.25 +2664 = 1834.75

z score =-1

raw score , y = -1* 3.17+5.45 = 2.28