Question

Anyone wanna give a shot at helping me answer an AP Stats question?

Answer 1

need to know more details

Answer 2

Alright let me take a second and type the question and part a

Answer 3

hope the site behaves enough to get thru it lol

Answer 4

"Human body temperatures takes through the ear are typically 0.5 degrees higher than body temperatures taken orally. Making this adjustment and using the 1992 Journal of the American Medical Association article that reports the average oral body temperature as 98.2 degrees, we will assume that a Normal model with an average of 98.7 degrees and a standard deviation of 0.7 degrees is appropriate for body temperatures taken through the ear.

a) An ear temperature of 97 degrees may indicate hypothermia (low body temperature). What percent of people have ear temperatures that may indicate hypothermia?


Okay so, I'm not sure how to get to the answer but I feel like somehow it's fairly simple. I do know that in a Normal Model,

68% of the data is 1 standard deviation away from the mean, 95% is 2 standard deviations away and 99.7% is 3 standard deviations away.

Answer 5

the empirical approximations prolly arent what they are looking for

do you know what a zscore, maybe called a standard score, is?

Answer 6

Yes we learned about z scores, it is like where the data is and if its above or below the mean or something like that

Answer 7

It tells you how many standard deviations something is from the mean

Answer 8

that is what we want to find to help us out. and is this work to be done by table, or by calculator?

Answer 9

Why would we use that if they're not asking for standard deviations?

Answer 10

you do realize that the empirical approximations that you mentioned tell us about probabilities base on standard deviations right?

Answer 11

if we know how many standard deviations something is from the mean, then we can determine probabilities from it

Answer 12

I don't understand a word you just said in that first part. I've never heard the word "empirical"

Answer 13

Oh I can find the z-score and go to that table thing to find my answer right?

Answer 14

well, in my textbook, many decades ago, the empirical rule is an approximation that states:

68% of the data is 1 standard deviation away from the mean, 95% is 2 standard deviations away and 99.7% is 3 standard deviations away.

Answer 15

yes, the z score allows us to use the tables

Answer 16

Oh our teacher just called it the 68-95-99.7 rule lol

Answer 17

So I know the equation for z score but I'm not sure which numbers I'd use for it

Answer 18

use the x, the mean, and the standard deviation stated in the problem of course

Answer 19

I know but I'm not sure which is which

Answer 20

I know the standard deviation of course

Answer 21

we will assume that a Normal model with an
average of 98.7 degrees and
a standard deviation of 0.7 degrees

a) An ear temperature of 97 degrees may indicate ...

Answer 22

Okay that's what I thought

so z=97-98.7/0.7 = -2.4 standard deviations below the mean

Answer 23

So I'll go to a z score of -2.4 on my table but then I'm not sure how to use the table from there

Answer 24

-2.428, a ztable tends to use 2 decimal points so id go with -2.43

Answer 25

My table just uses 1 so that's why I went -2.4

Answer 26

then thats fine too :)

Answer 27

So there's a whole row of numbers for -2.4 under each column so which do I know to use?

Answer 28

lol, the second decimal of course

Answer 29

What do you mean?

Answer 30

the side of the table should have the -2.4

the top of the table should have the rest of it: 0.03

Answer 31

Ohhh because the value was really like -2.43

Answer 32

where the row, -2.4, and the column, 0.03, cross at gives us a decimal value

Answer 33

yep

Answer 34

Ohhh so would that mean that 0.0075% of people have ear temperature that may indicate hypothermia?

Or do I still need to move the decimal over and make it .75%?

Answer 35

.0075 is a probability, a percentage needs to move the decimal

Answer 36

Alright so 0.75% of people

Answer 37

b) Find the interquartile range for ear temperatures

Could you help with part b as well?

Answer 38

I know the IQR is Q3-Q1

Answer 39

what are the percentages for an interquartile range?

Answer 40

25% and 75%

Answer 41

we need to find the z scores that relate to these percentages

Answer 42

look in the tables field for a value that is closest to .2500 and construct the z score form it

Answer 43

.2514 at -0.67

Answer 44

So is that our q1?

Answer 45

and .7514 at 0.68 is our Q3?

Answer 46

-0.67 seems fair to me yes

now, by symmetry Q1 and Q3 differ only by a sign.

chk z=.67 to see how close it is to .7500 if you have it with your table

Answer 47

.67 is .7486

Answer 48

now heres how we find the range we want.

the z score tells us how many deviations we are from the mean sooo

start with the mean, then add in a zscore multiple of the standard deviation

Answer 49

.674 is what i get from the wolf, which should give us plus/minus .67

Answer 50

Wait so which is it .67 or .68?

Answer 51

id go with -.67 and .67

but thats just my call

Answer 52

Okay and so that tells us the probability right?

Answer 53

no, we are trying to reconstruct an interval; not a probability

Answer 54

So now what do we do with those -.67 and .67 to find our Q3 and Q1?

Answer 55

we are given probabilities and work it backwards so to speak into the interval

Answer 56

since\[z=\frac{x-mean}{sd}\]we can work this to solve for x\[mean+ z(sd)=x\]
we need an interval from \(x_1~to~x_2\)

Answer 57

Okay and they are .67 and -.67 right? those are how many standard deviations the z score is above or below the mean

So how do we find the Q1 and Q3 now

Answer 58

I'm confused, an interval?

Answer 59

yes, an interval .... a range of values.

Answer 60

an interquartile range of values

Answer 61

Yes Q3 and Q1 right?

Answer 62

That is the info I need for this problem

Answer 63

yes, and ive posted how we can rework the zscore formula to find x (q1 or q3)

Answer 64

So do we use the avg temp and our new z scores and the same standard deviation to fint eh 2 different ones?

Answer 65

we know the mean, the z score, and the standard deviation .... we simply solve for x

Answer 66

yes

Answer 67

So Q1=98.2
and Q3 = 99.7
?

Answer 68

So our IQR is 99.7-98.2 = 1.5

Answer 69

our IQR is a range of values, an interval. and im getting 99.169 for q3

Answer 70

Yeah but IQR is calculated by Q3-Q1

Answer 71

this is where we get into some ambiguity with this term. but im pretty sure its not going to be the size of the difference ... but rather the actual interval that contains the values

Answer 72

youll have to check your material to be sure. since im not basing my understanding off of any material but my memories

Answer 73

i do see some sites defining it as q3-q1, so im not opposed to that as long as that is what your material is asking for

Answer 74

I think I'm right and it's not that big 99 number because of part C

"A new thermometer for the ear reports that it is more accurate than the ear thermometers currently on the market. If the avg temp stays the same (98.7) and the company reports an IQR of 0.5 degrees find the standard deviation for this new ear thermometer.

And yeah my notes say IQR=Q3-Q1

Answer 75

thats fine then :)

Answer 76

So now can you help with C? :P

Answer 77

well, we know the z score already from b so lets solve for the sd

\[z=\frac{x-mean}{sd}\implies sd=\frac{x-mean}{z}\]

Answer 78

let x be equal to mean + IQR/2 why?

Answer 79

which reduces us to sd = IQR/(2z)

Answer 80

I can't find a way to explain it but y=mean+IQR/2 makes sense

Answer 81

So we get s=0.5/2(z) except idk what z is

Answer 82

well, since the mean is half way in an interval that is IQR wide

the mean + IQR/2 gives us the largest x value to play with

Answer 83

\[sd=\frac{x-mean}{z}\]
\[sd=\frac{mean+IQR/2-mean}{z}\]
\[sd=\frac{IQR/2}{z}\]
\[sd=\frac{IQR}{2z}\]

Answer 84

Yes that I get and our IQR we were told is 0.5

Answer 85

and we determined the z score was .67 for the |q1| or |q3|

Answer 86

Oh we keep that ok

Answer 87

So sd = 0.37 degrees

yay

Answer 88

thats seems good to me

Answer 89

notice that this form give us IQR = sd(2z) which may be a simpler formula for IQR

Answer 90

Yay! Thanks so much, now I won't be really clueless on my test tomorrow on a question like this :)

Answer 91

good luck :)

Answer 92

Thanks!