2.) The equation yˆ=−8.74x2+50.57x+39.02 models the number of customers in a store x hours after opening.

Q 2 Options -Which number is the best estimate for the number of customers in the store 3 hours after opening?
A. 86
B. 47
C. 105
D. 112

3.The table shows the number of mobile cellular subscriptions per 100 people in Australia for different years.

Year 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011
Cellular subscriptions / 100 people 4 12 25 33 57 72 90 101 101 108
The quadratic regression equation using the years since 1990 for the input variable and the number of cellular subscriptions for the output variable is yˆ=−0.107x2+9.008x−28.851.

What is the predicted number of cellular subscriptions per 100 people in Australia in 2030?

Round your answer to the nearest whole number. Enter your answer in the box.

4. Determine the equation of the quadratic regression curve for the data.

x 0 1 2 3 4 5 6
y 4.1 −0.9 −3.9 −5.1 −4.1 −1.1 4.1
Enter your answer in the box

5.The table shows the average number of miles traveled per person in the United States for different years.

Year 1915 1925 1935 1945 1955 1965 1975 1985 1995 2005
Average miles traveled per person 194 1056 1796 1788 3650 4569 6147 7460 9099 10,181

Find the quadratic regression equation using the years since 1900 for the input variable and the average number of miles for the output variable. Round each coefficient in the equation to the nearest thousandth.

Using the regression equation with rounded values, what is the best estimate for the average number of miles traveled per person in 1961?

Question 5 options:

A. 4324


B. 1885


C. 4191


D. 2398

Answer:

2. The correct option is;

D. 112

3. The predicted number of cellular subscription per 100 people in Australia in 2030 is 217 subscriptions

4. The equation of the quadratic regression curve is y = x² - 6×x + 4.1

5. The closest option is;

C. 4191

Step-by-step explanation:

2. The equation for the number of customers in a store is presented as follows;

y = -8.74·x² + 50.57·x + 39.02

Where:

x = Number of hours after opening

∴ When x = 3 hours

y = -8.74×3² + 50.57×3 + 39.02 = 112.07 ≈ 112 people

The correct option is D. 112

3. The expression of the regression equation is presented as follows

y = -0.107·x² + 9.008·x +28.851

Where:

x = Number of years since 1990, hence

When x = 2030 - 1990 = 40 Years

Hence, the predicted number of cellular subscriptions per  people in 2030 is presented as follows;

y = -0.107×40² + 9.008×40 +28.851 = 217.971 Subscribers

To round down to the nearest whole number as y = 217 subscribers

4. The general form of a quadratic equation is presented as follows;

y = a·x² + b·x + c

When x = 0, y = 4.1, therefore;

4.1 =  a×0² + b×0 + c = c

∴ c = 4.1

When x = 1, y = -0.9, therefore;

-0.9 =  a×(1)² + b×1 + c = c

-0.9 =  a + b + 4.1

∴ a + b = -0.9 - 4.1 = -5.0

When x = 2, y = -3.9, therefore;

-3.9 =  a×(2)² + b×2 + c = c

-3.9 =  4·a + 2·b + 4.1

∴ 4·a + 2·b = -3.9 - 4.1 = -8.0

Thus we have two equations;

a + b = -5.0..................(1) and

4·a + 2·b = -8.0 ..........(2)

Multiply equation (1) by 2 and subtract it from equation (2), we have

4·a + 2·b - 2×(a + b) = *8.0 - (2 ×-5.0)

∴ 2·a = 2

a = 1

From equation (1,)  we have;

a + b = -5.0..................(1)

Therefore, where a = 1 we have, 1 + b = -5.0

Hence, b = -5.0  - 1 = -6.0

Therefore, the equation of the quadratic regression curve is presented as follows;

y = x² - 6×x + 4.1

5. The quadratic regression equation is found as follows;

The general form of a quadratic equation is presented as follows;

y = a·x² + b·x + c

When x = 55, y = 3650, therefore;

3650=  a×55² + b×55+ c................(1)

When x = 65, y = 4569, therefore;

4569=  a×65² + b×65+ c ...........(2)

When x = 75, y = 6147, therefore;

6147=  a×75² + b×75+ c............(3)

Solving the system of equation;

3650=  a×55² + b×55+ c................(1)

4569=  a×65² + b×65+ c ...........(2)

6147=  a×75² + b×75+ c............(3)

Subtracting equation (1) from (2), we obtain;

4569 - 3650 =  a×65² + b×65 + c - (a×55² + b×55 + c)

919 = 1200·a + 10·b...........(4)

Subtracting equation (2) from (3), we obtain;

6147 - 4569 =  a×75² + b×75 + c - (a×65² + b×65+ c)

Which gives;

1578 = 1400·a + 10·b...........(5)

Subtracting equation (4) from equation (3), we obtain;

1578 - 919 = 1400·a + 10·b - (1200·a + 10·b)

659 = 200·a

a = 659/200= 3.295

Substituting the value of a in equation (4), we have;

919 = 1200×3.295 + 10·b = 3954 + 10·b

∴ 10·b = 919 -3954 = -3035

b = -3035/10 = -303.5

Substituting the value of a and b in equation (1), we have;

3650 =  (3.295)×55² + (-303.5)×55+ c

3650 = -6725.125 + c

∴ c = 3650 - (- 6725.125)= 10375.125

Therefore, the quadratic regression equation is presented as follows;

y = 3.295·x - 303.5·b +10375.125

Hence, in 1961, x = 1961 - 1900 = 61, we have

y = 3.295×61² - 303.5×61 10375.125= 4122.32

Therefore, the closest option is C. 4191.

Answer:

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