Question

8. The table below contains the standardized heart rates (z-scores) of some babies and some adults. Use the table to answer the questions that follow.

Subjects:

Observations:

Babies	Heart Rate z-scores	Adults	Heart rate z-scores
Kiran	-1.53	Fran	0.66
Becca	-1.38	Pete	-1.95
Gavin	-0.59	Maria	-2.13
Dylan	2.52	Sarah	1.13
Emily	1.25	Mike	-0.64
Rachel	-0.04	Fred	0.56

The z-scores in each group here are not a complete set of z-scores so they will not add up to 0 and their std devn will not be 1. In this example, each set is just a partial set of z-scores.

(i) How many babies had heart rates that were below the baby mean?                            

(ii) How many adults had heart rates that were above the adult mean?                

(iii) Which baby’s heart rate was closest to the mean for babies?                        

(iv) Which adult’s heart rate was furthest from the mean for adults?                                 

(v) Which baby had the highest heart rate?                                                                    

(vi) Which adult had the lowest heart rate?                                                                     

(vii) Let us suppose that if a person’s heart rate is more than 2 standard deviations away from the mean for their age group, on either the fast or the slow side, then there is cause for concern. List the babies and adults for whom there is cause for concern.                                      

(viii) Of Emily and Pete,

who might have the greater need to have their heart rate monitored?                  

(ix) Of the baby with the highest heart rate and the adult with the highest

heart rate, which one was closer to the mean for their distribution?                                  

(x) What proportion of babies

have heart rates that are within 1 standard deviation of the mean?                                  

(xi) What proportion of adults

have heart rates that are within 1 standard deviation of the mean?                                  

(xii) Are the proportions of babies and adults who have heart rates

that are within 1 standard deviation of the mean equal?                                                  

Answer 1

(i)
For the mean, z score is 0.
Number of babies that had heart rates below the baby mean = Number of babies that z score less then 0 = 4

(ii)
For the mean, z score is 0.
Number of adults that had heart rates above the adults mean = Number of adults that z score greater then 0 = 3

(iii)
Heart rate was closest to the mean for babies will have Z score close to 0.
So, Rachel's heart rate was closest to the mean.

(iv)
Heart rate was farthest from the mean for adults will have largest absolute scores of Z.
So, Maria's heart rate was farthest from the mean because the absolute Z scores of Maria is largest among adults.

(v)
The baby with the highest heart rate will have highest Z scores.
Dylan have the highest heart rate.

(vi)
The adult with the lowest heart rate will have lowest Z scores.
Maria have the lowest heart rate.

(vii)
If a person’s heart rate is more than 2 standard deviations away from the mean, then their absolue Z score will be greater than 2.
Babies - Dylan
Adults - Maria

(viii) Pete have the greater need to have their heart rate monitored, since the absolute z value of Pete is greater and more away from the normal rate.

(ix)
Baby with the highest heart rate is Dylan with z score of 2.52
Adult with the highest heart rate is Sarah with z score of 1.13
Since z score of Sarah is lesser, it is more close to the mean.

(x)
Proportion of babies have heart rates that are within 1 standard deviation of the mean = Number of babies with absolute z score less than 1 / Total number of babies
= 2 / 6 = 1/3

(xi)
Proportion of adults have heart rates that are within 1 standard deviation of the mean = Number of adults with absolute z score less than 1 / Total number of babies
= 3 / 6 = 1/2

(xii)
No, the proportions of babies and adults who have heart rates that are within 1 standard deviation of the mean are not equal