A vending machine is designed to dispense 8 ounces of coffee

Question

7.51 8.22 7.86 8.36
8.09 7.83 8.30 8.01
7.73 8.25 7.96 8.53

A vending machine is designed to dispense 8 ounces of coffee into a cup. After a test that recorded the number of ounces of coffee in each of 1000 cups dispensed by the vending machine, the 12 listed amounts, in ounces, were selected from the data above. If the 1000 recorded amounts have a mean of 8.1 ounces and a standard deviation of 0.3 ounces, how many of the 12 listed amounts are within 1.5 standard deviation of the mean?

A. Four
B. Six
C. Nine
D. Ten
E. Eleven

102037

Bunuel · Accepted Answer

"Within 1.5 standard deviation of the mean" - means in the range

{mean - 1.5*sd; mean + 1.5*sd} =

= {8.1 - 1.5*0.3; 8.1 + 1.5*0.3} =

= {7.65; 8.55} .

From the 12 listed amounts, only one (7.51) is out of this range and 11 are within this range.

Answer: E.

Bunuel · Answer

Please check the post just above yours: a-vending-machine-is-designed-to-dispense-8-ounces-of-coffee-93351.html#p1142674 Similar questions to practice: arithmetic-mean-and-standard-deviation-of-a-certain-normal-104117.html 70-75-80-85-90-105-105-130-130-130-the-list-shown-consist-of-100361.html for-a-certain-exam-a-score-of-58-was-2-standard-deviations-b-128661.html a-certain-characteristic-in-a-large-population-has-a-143982.html a-vending-machine-is-designed-to-dispense-8-ounces-of-coffee-93351.html the-residents-of-town-x-participated-in-a-survey-83362.html the-standard-deviation-of-a-normal-distribution-of-data-is-99221.html the-mean-and-the-standard-deviation-of-the-8-numbers-shown-98248.html the-lifetime-of-all-the-batteries-produced-by-a-certain-comp-101472.html

BrentGMATPrepNow · Answer

If the Standard Deviation is 0.3 ounces, then 0.3 ounces represents  1 unit of standard deviation . 
Similarly, 0.6 ounces represents  2 units of standard deviation , 
0.15 ounces represents  0.5 units of standard deviation , and so on.

If the mean is 8.1 ounces, then we say that 8.4 ounces is 1 unit of standard deviation  above  the mean (since 8.1 + 0.3 = 8.4), and we say that 7.8 ounces is 1 unit of standard deviation  below  the mean (since 8.1 - 0.3 = 7.8)

We want to know how many measurements in the list are within  1.5 standard deviations of the mean

Well, using the above logic,  0.45 represents 1.5 units of standard deviation .

So, 1.5 units of standard deviation  below  the mean equals 7.65 (8.1 -  0.45  = 7.65)
Similarly, 1.5 units of standard deviation  above the mean equals 8.55 (8.1 +  0.45  = 8.55)

So, any measurement that is between 7.65 ounces and 8.55 will be within 1.5 standard deviations of the mean.

In the given list of measurements, the following meet this requirement:

7.51  8.22 7.86 8.36  
 8.09 7.83 8.30 8.01  
 7.73 8.25 7.96 8.53

So the answer is 11 (E)

Cheers,
Brent

MICKEYXITIN · Answer

Source: GMAT Prep

"Within 1.5 standard deviation of the mean" - means in the range  {mean-1.5*sd; mean+1.5*sd} = {8.1-1.5*0.3; 8.1+1.5*0.3} = {7.65; 8.55} .

From the 12 listed amounts, only one (7.51) is out of this range and 11 are within this range.

Answer: E. 
 hi Bunuel,
 I have not get clear why you take: 1.5*0.3
did i miss something? Please explain for me

Bunuel · Answer

We are given that SD=0.3 and mean=8.1. Question: how many of the 12 listed amounts are  within  1.5  standard deviation  of the mean?

1.5 standard deviation equals to 1.5*SD=1.5*0.3=0.45, so "within  1.5  standard deviation of them mean" - means in the range  {8.1-0.45; 8.1+0.45} = {7.65; 8.55} .

Hope it's clear.

NON · Answer

Does the range include specifically numbers 7.65 and 8.55?

Bunuel · Answer

Luckily we don't need to worry about it for this question, though I'd say yes, the range has to be inclusive.

imhimanshu · Answer

Hello Bunuel,
Would you please clear one doubt. 
The question states that "how many of the 12 listed amounts are   within 1.5 standard deviation  of the mean? In this case, do we need to consider boundary values as well such as (7.65) and (8.55). 
I mean, If the question had 7.65 value, then would we have considered 7.65 value ? i.e  7.65<= std deviation <= 8.55  is the range
Another doubt:
If the question had asked
Question: how many of the 12 listed amounts are  between 1.5 standard deviation  of the mean?, and the answer choice had 7.65 as the value

Then, do we need to consider 7.65 value while answering ? I believe, not and  7.65< std deviation < 8.55   will be the range
Please provide your comments.

pacifist85 · Answer

So:
M= 8.1
SD= 0.3
1.5 SD= 1 SD plus 1/2 SD, which is 0.3+0.15=0.45 SD.

Lets add and subtract 3SD or 0.45 from the M or 8.1:
8.1 - 0.45 = 7.65
8.1 + 0.45 = 8.55

So, we are looking for numbers in this range 7.65<numbers<8.55.

These are 11 of the 12 numbers,  ANS E .

GMATinsight · Answer

Mean = 8.1
Standard Deviation = 0.3

within 1.5 standard deviation of the mean

i.e. 1.5 standard deviation ABOVE the mean = 8.1+1.5*0.3 = 8.55
and 1.5 standard deviation BELOW the mean = 8.1-1.5*0.3 = 7.65

Except first value 7.51, all other values are within the range hence, 11 values fall within 1.5 SD from Mean

Answer: Option E

Ekland · Answer

Thanks  GMATinsight  but why did you have multiply 1.5 with 0.3?

GMATinsight · Answer

1.5 standard deviation   means   1.5*Standard Deviation

i.e. 1.5 standard deviation   ABOVE   the mean = 8.1+1.5*0.3 = 8.55
and 1.5 standard deviation   BELOW   the mean = 8.1-1.5*0.3 = 7.65

I hope this helps!

shettytarun · Answer

Hi Bunuel, I am still not clear on this problem.

since 8.1 is the mean, i also assume 0.3 is the SD on this mean.

and if the questions asks 1.5 SD of the mean, doesn't it translate to 0.75 (1.5/2) either side of 1.5.

Why do we multiply the 1st SD and the 2nd.

Please help

Bunuel · Answer

I tried to explain this here: https://gmatclub.com/forum/a-vending-ma ... ml#p812247 We are given that SD=0.3 and mean=8.1. Question: how many of the 12 listed amounts are within 1.5 standard deviation of the mean? 1.5 standard deviation equals to 1.5*SD=1.5*0.3=0.45, so "within 1.5 standard deviation of them mean" - means in the range {8.1-0.45; 8.1+0.45} = {7.65; 8.55} . Hope it's clear.

JeffTargetTestPrep · Answer

We are given 12 listed amounts of coffee that we can put in order from least to greatest.
 
7.51, 7.73, 7.83, 7.86, 7.96, 8.01, 8.09, 8.22, 8.25, 8.30, 8.36, 8.53
 
We need to determine how many of the listed amounts are within 1.5 standard deviations of the mean. Since the standard deviation is 0.3, 1.5 standard deviations is: 1.5 x 0.3 = 0.45.
 
1.5 standard deviations above the mean is 8.1 + 0.45 = 8.55
 
1.5 standard deviations below the mean is 8.1 – 0.45 = 7.65
 
We see that all the values are between 7.65 and 8.55 except 7.51. Thus, 11 values are within 1.5 standard deviation of the mean.
 
Answer: E

GmatPoint · Answer

Mean = 8.1
Range =  
Range =

Out of the 12 observations, only 7.51 is out of the range.
Thus, 11 of the 12 listed amounts are within a 1.5 standard deviation of the mean.
The correct option is E.

nikitathegreat · Answer

We have been given Mean and SD for 1000 records. Does the same SD and Mean will apply to the 12 records? or do we have to take out separate mean for 1000 records?

Bunuel · Answer

There are 1000 recorded amounts, which have a mean of 8.1 ounces and a standard deviation of 0.3 ounces. The mean and the standard deviation of the 12 listed amounts are different: 8.05417 and 0.280994, respectively. However, for this question, the mean and the standard deviation of the 12 listed amounts do not matter.

bumpbot · Answer

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A vending machine is designed to dispense 8 ounces of coffee

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