Question

The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about   36   and a standard deviation of about   6 . Suppose that   20   individuals are randomly chosen.

Let   X¯=   average percent of fat calories consumed for a sample of size   20 .

Enter numbers as integers or fractions in "p/q" form, or as decimals accurate to nearest   0.01 .

1. (.20)   X¯∼    (pick one) ENBP    (  , ) .
2. 
   
3. (.20)   Use the mean and standard deviation of   X¯   to determine the   z   value for   X¯=35 .
   

   z=    .

4. 
   
5. (.20)   Illustrate   P(X¯<35)   as an area by adjusting the slider along the horizontal axis to control the   z   value.
   

   -3

   -2

   -1

   0

   1

   2

   -0.2

   -0.1

   0.1

   0.2

   0.300

   0.4

   –  o  +  ←  ↑  ↓  →

   z=

   0.00

   Fill left    Fill right

   
   
   
6. 
   
7. (.20)   What is the probability that the average fat consumption for   20   people in America is less than   35 ?
   

   P(X¯<35)=

8. 
   
9. (.20)  Find the first quartile for the average percent of fat calories.
   

   Q1=   

Answer 1

a) Since the distribution is normal hence using Central limit theorem

$\bar{X}\sim 36$

b) it has no question,

c) At $\bar{X}=35$

where population mean =36 and population standard deviation =6 and n, no of sample =20

using Z formula as

$Z=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}$

Z= -0.7454

e) The curve as for P(X¯<35)

g) P(X¯<35) calculated using Z value at X¯=35 , which is calculated above as -0.745

hence using Z table

P(X¯<35)=0.2266

i) At Quartile 1 the percentile is 25th hence at Quartile 1 that is at P=0.25, from Z table again

Z at P=0.25 is -0.675 hence by Z formula

$Z=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}$

first quartile for the average percent of fat calories.

Q1=   35.09