Question

score on last try: 0 of 1 pts. see details for more. at least one scored part is incorrect. jump to first changable incorrect part. > next question you can retry this question below statistics students at a local college sampled 9 textbooks in the school bookstore, and recorded the number of pages in each textbook along with its cost. the data is shown below, number of pages (x) | cost(y) 937 | 127.18 905 | 129.7 786 | 110.04 735 | 107.9 671 | 86.94 478 | 67.92 804 | 114.56 338 | 47.32 925 | 126.5 a student calculates a linear model y = . (please show your answers to two decimal places) use the model above to estimate the cost when number of pages is 501 cost = $ (please show your answer to 2 decimal places.) enter an integer or decimal number more.. question help: message instructor

Explanation:

Step1: Calculate necessary sums

First, we need to find the sums of \( x \), \( y \), \( xy \), and \( x^2 \) for the given data.
The data points are:
\( (937, 127.18) \), \( (905, 129.7) \), \( (786, 110.04) \), \( (735, 107.9) \), \( (671, 86.94) \), \( (478, 67.92) \), \( (804, 114.56) \), \( (338, 47.32) \), \( (925, 126.5) \)

\( n = 9 \)

\( \sum x = 937 + 905 + 786 + 735 + 671 + 478 + 804 + 338 + 925 = 6579 \)

\( \sum y = 127.18 + 129.7 + 110.04 + 107.9 + 86.94 + 67.92 + 114.56 + 47.32 + 126.5 = 918.06 \)

\( \sum xy = (937\times127.18) + (905\times129.7) + (786\times110.04) + (735\times107.9) + (671\times86.94) + (478\times67.92) + (804\times114.56) + (338\times47.32) + (925\times126.5) \)
Calculating each term:
\( 937\times127.18 = 119167.66 \)
\( 905\times129.7 = 117378.5 \)
\( 786\times110.04 = 86501.44 \)
\( 735\times107.9 = 79306.5 \)
\( 671\times86.94 = 58336.74 \)
\( 478\times67.92 = 32465.76 \)
\( 804\times114.56 = 92006.24 \)
\( 338\times47.32 = 15994.16 \)
\( 925\times126.5 = 117012.5 \)
\( \sum xy = 119167.66 + 117378.5 + 86501.44 + 79306.5 + 58336.74 + 32465.76 + 92006.24 + 15994.16 + 117012.5 = 717179.5 \)

\( \sum x^2 = 937^2 + 905^2 + 786^2 + 735^2 + 671^2 + 478^2 + 804^2 + 338^2 + 925^2 \)
Calculating each term:
\( 937^2 = 878969 \)
\( 905^2 = 819025 \)
\( 786^2 = 617796 \)
\( 735^2 = 540225 \)
\( 671^2 = 450241 \)
\( 478^2 = 228484 \)
\( 804^2 = 646416 \)
\( 338^2 = 114244 \)
\( 925^2 = 855625 \)
\( \sum x^2 = 878969 + 819025 + 617796 + 540225 + 450241 + 228484 + 646416 + 114244 + 855625 = 5151025 \)

Step2: Calculate slope \( m \) and intercept \( b \)

The formula for the slope \( m \) of the linear regression line is:
\( m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)

Substitute the values:
\( m = \frac{9\times717179.5 - 6579\times918.06}{9\times5151025 - (6579)^2} \)

First, calculate numerator:
\( 9\times717179.5 = 6454615.5 \)
\( 6579\times918.06 = 6579\times918.06 \approx 6579\times900 = 5921100 \), \( 6579\times18.06 = 118816.74 \), total \( 5921100 + 118816.74 = 6039916.74 \)
Numerator \( = 6454615.5 - 6039916.74 = 414698.76 \)

Denominator:
\( 9\times5151025 = 46359225 \)
\( (6579)^2 = 6579\times6579 = 43283241 \)
Denominator \( = 46359225 - 43283241 = 3075984 \)

\( m = \frac{414698.76}{3075984} \approx 0.1348 \approx 0.13 \) (Wait, let's recalculate more accurately)
Wait, maybe I made a mistake in calculation. Let's recalculate \( \sum x \) and \( \sum y \) again.

Wait, \( \sum x = 937 + 905 = 1842; +786=2628; +735=3363; +671=4034; +478=4512; +804=5316; +338=5654; +925=6579 \). Correct.

\( \sum y = 127.18 + 129.7 = 256.88; +110.04=366.92; +107.9=474.82; +86.94=561.76; +67.92=629.68; +114.56=744.24; +47.32=791.56; +126.5=918.06 \). Correct.

\( \sum xy \): Let's recalculate one term: 338*47.32: 300*47.32=14196, 38*47.32=1808.16, total 14196+1808.16=16004.16. Wait, earlier I had 15994.16, that's a mistake. So correct \( \sum xy \):

937*127.18=119167.66

905*129.7=117378.5

786*110.04=86501.44

735*107.9=79306.5

671*86.94=58336.74

478*67.92=32465.76

804*114.56=92006.24

338*47.32=16004.16 (corrected)

925*126.5=117012.5

Now sum these:

119167.66 + 117378.5 = 236546.16

+86501.44 = 323047.6

+79306.5 = 402354.1

+58336.74 = 460690.84

+32465.76 = 493156.6

+92006.24 = 585162.84

+16004.16 = 601167

+117012.5 = 718179.5

Ah, I see, earlier I had 717179.5, but it's 718179.5. That was the mistake.

Now recalculate numerator:

\( 9\times718179.5 = 6463615.5 \)

\( 6579\times918.06 \): Let's calculate 6579*918.06

First, 6579*900 = 5921100

6579*18.06 = 6579*(18 + 0.06) = 6579*18 + 6579*0.06 = 118422 + 394.74 = 118816.74

So total \( 5921100 + 118816.74 = 6039916.74 \)

Numerator = 6463615.5 - 6039916.74 = 423698.76

Denominator:

\( 9\times5151025 = 46359225 \)

\( (6579)^2 = 6579*6579 \). Let's calculate 6500^2 = 42250000, 79^2=6241, 2*6500*79=1027000, so (6500+79)^2=42250000 + 1027000 + 6241=43283241. Correct.

Denominator = 46359225 - 43283241 = 3075984

Now \( m = \frac{423698.76}{3075984} \approx 0.1377 \approx 0.14 \) (Wait, let's do division: 423698.76 ÷ 3075984 ≈ 0.1377, so approximately 0.14)

Now calculate the intercept \( b \):

\( b = \frac{\sum y - m\sum x}{n} \)

\( \sum y = 918.06 \), \( m\sum x = 0.1377\times6579 \approx 0.1377\times6579 \approx 906.93 \)

\( b = \frac{918.06 - 906.93}{9} = \frac{11.13}{9} \approx 1.24 \)

Wait, that can't be right. Wait, maybe my calculation of \( m \) is wrong. Let's use a better approach. Let's use the formula for linear regression:

The linear model is \( y = mx + b \), where \( m = \frac{S_{xy}}{S_{xx}} \), \( b = \bar{y} - m\bar{x} \)

Where \( S_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n} \)

\( S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n} \)

\( \bar{x} = \frac{\sum x}{n} \), \( \bar{y} = \frac{\sum y}{n} \)

Let's recalculate using these formulas.

\( \bar{x} = \frac{6579}{9} = 731 \)

\( \bar{y} = \frac{918.06}{9} = 102.006666... \approx 102.01 \)

\( S_{xy} = 718179.5 - \frac{6579\times918.06}{9} \)

Calculate \( \frac{6579\times918.06}{9} = \frac{6039916.74}{9} = 671101.86 \)

\( S_{xy} = 718179.5 - 671101.86 = 47077.64 \)

\( S_{xx} = 5151025 - \frac{(6579)^2}{9} = 5151025 - \frac{43283241}{9} = 5151025 - 4809249 = 341776 \)

Now \( m = \frac{47077.64}{341776} \approx 0.1377 \approx 0.14 \)

\( b = \bar{y} - m\bar{x} = 102.0067 - 0.1377\times731 \)

Calculate \( 0.1377\times731 \approx 0.1377\times700 = 96.39, 0.1377\times31=4.2687, total 96.39+4.2687=100.6587 \)

\( b = 102.0067 - 100.6587 = 1.348 \approx 1.35 \)

Wait, but let's check with actual data. Let's take a point, say (338, 47.32). If \( y = 0.14x + 1.35 \), then \( y = 0.14*338 + 1.35 = 47.32 + 1.35 = 48.67 \), but actual y is 47.32. Close, but not exact. Maybe my calculation of \( \sum xy \) is still wrong.

Wait, let's recalculate \( \sum xy \) correctly:

1. (937, 127.18): 937*127.18 = 937*127 + 937*0.18 = 118999 + 168.66 = 119167.66 ✔️

2. (905, 129.7): 905*129.7 = 905*(130 - 0.3) = 117650 - 271.5 = 117378.5 ✔️

3. (786, 110.04): 786*110.04 = 786*110 + 786*0.04 = 86460 + 31.44 = 86491.44? Wait, earlier I had 86501.44. Oh! Here's the mistake. 786*110.04: 786*110 = 86460, 786*0.04 = 31.44, so total 864

Answer:

Step1: Calculate necessary sums

First, we need to find the sums of \( x \), \( y \), \( xy \), and \( x^2 \) for the given data.
The data points are:
\( (937, 127.18) \), \( (905, 129.7) \), \( (786, 110.04) \), \( (735, 107.9) \), \( (671, 86.94) \), \( (478, 67.92) \), \( (804, 114.56) \), \( (338, 47.32) \), \( (925, 126.5) \)

\( n = 9 \)

\( \sum x = 937 + 905 + 786 + 735 + 671 + 478 + 804 + 338 + 925 = 6579 \)

\( \sum y = 127.18 + 129.7 + 110.04 + 107.9 + 86.94 + 67.92 + 114.56 + 47.32 + 126.5 = 918.06 \)

\( \sum xy = (937\times127.18) + (905\times129.7) + (786\times110.04) + (735\times107.9) + (671\times86.94) + (478\times67.92) + (804\times114.56) + (338\times47.32) + (925\times126.5) \)
Calculating each term:
\( 937\times127.18 = 119167.66 \)
\( 905\times129.7 = 117378.5 \)
\( 786\times110.04 = 86501.44 \)
\( 735\times107.9 = 79306.5 \)
\( 671\times86.94 = 58336.74 \)
\( 478\times67.92 = 32465.76 \)
\( 804\times114.56 = 92006.24 \)
\( 338\times47.32 = 15994.16 \)
\( 925\times126.5 = 117012.5 \)
\( \sum xy = 119167.66 + 117378.5 + 86501.44 + 79306.5 + 58336.74 + 32465.76 + 92006.24 + 15994.16 + 117012.5 = 717179.5 \)

\( \sum x^2 = 937^2 + 905^2 + 786^2 + 735^2 + 671^2 + 478^2 + 804^2 + 338^2 + 925^2 \)
Calculating each term:
\( 937^2 = 878969 \)
\( 905^2 = 819025 \)
\( 786^2 = 617796 \)
\( 735^2 = 540225 \)
\( 671^2 = 450241 \)
\( 478^2 = 228484 \)
\( 804^2 = 646416 \)
\( 338^2 = 114244 \)
\( 925^2 = 855625 \)
\( \sum x^2 = 878969 + 819025 + 617796 + 540225 + 450241 + 228484 + 646416 + 114244 + 855625 = 5151025 \)

Step2: Calculate slope \( m \) and intercept \( b \)

The formula for the slope \( m \) of the linear regression line is:
\( m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)

Substitute the values:
\( m = \frac{9\times717179.5 - 6579\times918.06}{9\times5151025 - (6579)^2} \)

First, calculate numerator:
\( 9\times717179.5 = 6454615.5 \)
\( 6579\times918.06 = 6579\times918.06 \approx 6579\times900 = 5921100 \), \( 6579\times18.06 = 118816.74 \), total \( 5921100 + 118816.74 = 6039916.74 \)
Numerator \( = 6454615.5 - 6039916.74 = 414698.76 \)

Denominator:
\( 9\times5151025 = 46359225 \)
\( (6579)^2 = 6579\times6579 = 43283241 \)
Denominator \( = 46359225 - 43283241 = 3075984 \)

\( m = \frac{414698.76}{3075984} \approx 0.1348 \approx 0.13 \) (Wait, let's recalculate more accurately)
Wait, maybe I made a mistake in calculation. Let's recalculate \( \sum x \) and \( \sum y \) again.

Wait, \( \sum x = 937 + 905 = 1842; +786=2628; +735=3363; +671=4034; +478=4512; +804=5316; +338=5654; +925=6579 \). Correct.

\( \sum y = 127.18 + 129.7 = 256.88; +110.04=366.92; +107.9=474.82; +86.94=561.76; +67.92=629.68; +114.56=744.24; +47.32=791.56; +126.5=918.06 \). Correct.

\( \sum xy \): Let's recalculate one term: 33847.32: 30047.32=14196, 38*47.32=1808.16, total 14196+1808.16=16004.16. Wait, earlier I had 15994.16, that's a mistake. So correct \( \sum xy \):

937*127.18=119167.66

905*129.7=117378.5

786*110.04=86501.44

735*107.9=79306.5

671*86.94=58336.74

478*67.92=32465.76

804*114.56=92006.24

338*47.32=16004.16 (corrected)

925*126.5=117012.5

Now sum these:

119167.66 + 117378.5 = 236546.16

+86501.44 = 323047.6

+79306.5 = 402354.1

+58336.74 = 460690.84

+32465.76 = 493156.6

+92006.24 = 585162.84

+16004.16 = 601167

+117012.5 = 718179.5

Ah, I see, earlier I had 717179.5, but it's 718179.5. That was the mistake.

Now recalculate numerator:

\( 9\times718179.5 = 6463615.5 \)

\( 6579\times918.06 \): Let's calculate 6579*918.06

First, 6579*900 = 5921100

657918.06 = 6579(18 + 0.06) = 657918 + 65790.06 = 118422 + 394.74 = 118816.74

So total \( 5921100 + 118816.74 = 6039916.74 \)

Numerator = 6463615.5 - 6039916.74 = 423698.76

Denominator:

\( 9\times5151025 = 46359225 \)

\( (6579)^2 = 65796579 \). Let's calculate 6500^2 = 42250000, 79^2=6241, 26500*79=1027000, so (6500+79)^2=42250000 + 1027000 + 6241=43283241. Correct.

Denominator = 46359225 - 43283241 = 3075984

Now \( m = \frac{423698.76}{3075984} \approx 0.1377 \approx 0.14 \) (Wait, let's do division: 423698.76 ÷ 3075984 ≈ 0.1377, so approximately 0.14)

Now calculate the intercept \( b \):

\( b = \frac{\sum y - m\sum x}{n} \)

\( \sum y = 918.06 \), \( m\sum x = 0.1377\times6579 \approx 0.1377\times6579 \approx 906.93 \)

\( b = \frac{918.06 - 906.93}{9} = \frac{11.13}{9} \approx 1.24 \)

Wait, that can't be right. Wait, maybe my calculation of \( m \) is wrong. Let's use a better approach. Let's use the formula for linear regression:

The linear model is \( y = mx + b \), where \( m = \frac{S_{xy}}{S_{xx}} \), \( b = \bar{y} - m\bar{x} \)

Where \( S_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n} \)

\( S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n} \)

\( \bar{x} = \frac{\sum x}{n} \), \( \bar{y} = \frac{\sum y}{n} \)

Let's recalculate using these formulas.

\( \bar{x} = \frac{6579}{9} = 731 \)

\( \bar{y} = \frac{918.06}{9} = 102.006666... \approx 102.01 \)

\( S_{xy} = 718179.5 - \frac{6579\times918.06}{9} \)

Calculate \( \frac{6579\times918.06}{9} = \frac{6039916.74}{9} = 671101.86 \)

\( S_{xy} = 718179.5 - 671101.86 = 47077.64 \)

\( S_{xx} = 5151025 - \frac{(6579)^2}{9} = 5151025 - \frac{43283241}{9} = 5151025 - 4809249 = 341776 \)

Now \( m = \frac{47077.64}{341776} \approx 0.1377 \approx 0.14 \)

\( b = \bar{y} - m\bar{x} = 102.0067 - 0.1377\times731 \)

Calculate \( 0.1377\times731 \approx 0.1377\times700 = 96.39, 0.1377\times31=4.2687, total 96.39+4.2687=100.6587 \)

\( b = 102.0067 - 100.6587 = 1.348 \approx 1.35 \)

Wait, but let's check with actual data. Let's take a point, say (338, 47.32). If \( y = 0.14x + 1.35 \), then \( y = 0.14*338 + 1.35 = 47.32 + 1.35 = 48.67 \), but actual y is 47.32. Close, but not exact. Maybe my calculation of \( \sum xy \) is still wrong.

Wait, let's recalculate \( \sum xy \) correctly:

1. (937, 127.18): 937127.18 = 937127 + 937*0.18 = 118999 + 168.66 = 119167.66 ✔️

1. (905, 129.7): 905129.7 = 905(130 - 0.3) = 117650 - 271.5 = 117378.5 ✔️

1. (786, 110.04): 786110.04 = 786110 + 7860.04 = 86460 + 31.44 = 86491.44? Wait, earlier I had 86501.44. Oh! Here's the mistake. 786110.04: 786110 = 86460, 7860.04 = 31.44, so total 864