Mathematics, 30.08.2019 16:40 KillerSteamcar

Find the population standard deviation (σ not s) for the data set: {75,80,90,95,75,85,55,80,90,95}
round your answer to the nearest 10th.

Answers: 1

Show answers

Answers

There are a few steps involved which will follow this basic outline
Step 1) Find the sample mean (xbar)
Step 2) Subtract the xbar value from each data point. This list is the set of deviations from the mean.
Step 3) Square each result found in step 2. This list is the set of squared deviations from the mean.
Step 4) Add up the results found in step 3. This value is known as the sum of the squared deviations
Step 5) Divide the value found in the previous step by n = 10 to yield the population variance
Step 6) Apply the square root to the population variance to get the population standard deviation

Step 1) Find the sample mean xbar
Add up the values. Divide the sum by 10 (since there are n = 10 values)
xbar = (Sum of the values)/10
xbar = (75+80+90+95+75+85+55+80+90+95)/10
xbar = 820/10
xbar = 82
The sample mean is xbar = 82

Step 2) Subtract the sample mean from EVERY data value to form a new list
75-82 = -7
80-82 = -2
90-82 = 8
95-82 = 13
75-82 = -7
85-82 = 3
55-82 = -27
80-82 = -2
90-82 = 8
95-82 = 13
The differences are: -7, -2, 8, 13, -7, 3, -27, -2, 8, 13
This new list {-7, -2, 8, 13, -7, 3, -27, -2, 8, 13} is the list of deviations from the mean
In other words, it's the list of differences from the mean.

Step 3) Square each result found in step 2 to get the list of squared deviations
(-7)^2 = 49
(-2)^2 = 4
(8)^2 = 64
(13)^2 = 169
(-7)^2 = 49
(3)^2 = 9
(-27)^2 = 729
(-2)^2 = 4
(8)^2 = 64
(13)^2 = 169
The results from above are: 49, 4, 64, 169, 49, 9, 729, 4, 64, 169
The list of squared deviations from the mean is {49, 4, 64, 169, 49, 9, 729, 4, 64, 169}

Step 4) Add up each result found in the previous step to get the sum of the squared deviations.
49+4+64+169+49+9+729+4+64+169 = 1310
The sum of the squared deviations is equal to 1310

Step 5) Divide the sum of the squared deviations (1310) by n = 10 to get 1310/10 = 131
The population variance is 131

Step 6) Take the square root of population variance 131 to get sqrt(131) = 11.4455231422596
The population standard deviation is 11.4455231422596
Round that to the nearest tenth to get 11.4

Final 11.4

Answer from: Quest

part a

answer: a reasonable domain is $0 \le n \le 10$ where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

=====================================

part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

=====================================

part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).

Answer from: Quest

x = -4

step-by-step explanation:

- 20 - 4(x + 3) = 4x

=> -20 - 4x - 12 = 4x

=> - 32 -4x = 4x

=> -32 = 4x + 4x

=> - 32 = 8x

=> -32/8 = x

=> -4 = x