Math  /  Data & Statistics

QuestionWhen using population size as the explanatory variable, x , and broadband subscribers as the response variable, y , for data on the number of individuals in a country with broadband access and the population size for 38 nations, the regression equation is y^=4,995,453+0.0298x\hat{y}=4,995,453+0.0298 x. a. Interpret the slope of the regression equation. Is the association positive or negative? Explain what this means. b. Predict broadband subscribers at the (i) population size 7,010,0967,010,096, (ii) population size 1,109,575,9041,109,575,904. c. For one nation, y=71,610,000y=71,610,000, and x=308,536,233x=308,536,233. Find the predicted broadband use and the residual for this nation. Interpret the value of this residual. a. Since the association is positive, the slope means that as the \square increases by 1 unit, the number of broadband subscribers tends to \square increase \square 0.0298 (Type an integer or a decimal.) b. (i) The predicted broadband subscribers for population size 7,010,096 is 5,204,3545,204,354. (Round to the nearest whole number as needed.) (ii) The predicted broadband subscribers for population size 1,109,575,904 is 38,060,815. (Round to the nearest whole number as needed.) c. The predicted broadband subscribers for the nation is (Round to the nearest whole number as needed.)

Studdy Solution

STEP 1

What is this asking? We're looking at how population size relates to the number of broadband subscribers and making predictions based on that relationship! Watch out! Don't mix up the population size (xx) and broadband subscribers (yy).
Also, keep track of those large numbers carefully!

STEP 2

1. Interpret the Slope
2. Predict Broadband Subscribers
3. Calculate Predicted Use and Residual

STEP 3

The slope of our regression equation, y^=4,995,453+0.0298x\hat{y} = 4,995,453 + 0.0298x, is **0.0298**.
This tells us how much the predicted number of broadband subscribers (y^\hat{y}) changes for every increase of 1 in the population size (xx).

STEP 4

Since the slope is **positive**, there's a *positive association*.
This means as the population size *increases*, the number of broadband subscribers tends to *increase* as well!
Specifically, for every additional person in the population, the predicted number of broadband subscribers increases by **0.0298**.

STEP 5

We **plug in** x=7,010,096x = 7,010,096 into our equation: y^=4,995,453+0.02987,010,096\hat{y} = 4,995,453 + 0.0298 \cdot 7,010,096.

STEP 6

Calculating this gives us y^=4,995,453+208,999.7008=5,204,452.7008\hat{y} = 4,995,453 + 208,999.7008 = 5,204,452.7008.

STEP 7

Rounding to the nearest whole number, we get **5,204,453** predicted broadband subscribers.

STEP 8

Now we **plug in** x=1,109,575,904x = 1,109,575,904: y^=4,995,453+0.02981,109,575,904\hat{y} = 4,995,453 + 0.0298 \cdot 1,109,575,904.

STEP 9

This gives us y^=4,995,453+33,062,360.5192=38,057,813.5192\hat{y} = 4,995,453 + 33,062,360.5192 = 38,057,813.5192.

STEP 10

Rounding to the nearest whole number, we predict **38,057,814** broadband subscribers.

STEP 11

For x=308,536,233x = 308,536,233, we **plug it in**: y^=4,995,453+0.0298308,536,233\hat{y} = 4,995,453 + 0.0298 \cdot 308,536,233.

STEP 12

Calculating the predicted value: y^=4,995,453+9,193,746.2734=14,189,199.2734\hat{y} = 4,995,453 + 9,193,746.2734 = 14,189,199.2734.

STEP 13

Rounding gives us **14,189,200** predicted broadband subscribers.

STEP 14

The *residual* is the difference between the *actual* value of yy and the *predicted* value (y^\hat{y}).
We're given y=71,610,000y = 71,610,000.

STEP 15

So, the residual is 71,610,00014,189,200=57,420,80071,610,000 - 14,189,200 = 57,420,800.

STEP 16

This **positive residual** of **57,420,800** means the actual number of broadband subscribers for this nation is *much higher* than what our model predicted based on its population size.

STEP 17

a. As the *population size* increases by 1 unit, the number of broadband subscribers tends to increase by **0.0298**.
The association is *positive*. b. (i) **5,204,453** (ii) **38,057,814** c. The predicted broadband use is **14,189,200**.
The residual is **57,420,800**, meaning this nation has significantly more broadband subscribers than predicted.

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