Question

below are the overhead widths (cm) of seals measured from photographs and weights (kg) of the seals. find the regression equation, letting the overhead width be the predictor (x) variable. find the best predicted weight of a seal if the overhead width measured from a photograph is 1.7 cm, using the regression equation. can the prediction be correct? if not, what is wrong? use a significance level of 0.05.

overhead width (cm) | 7.1 | 7.4 | 9.8 | 9.4 | 8.9 | 8.5
weight (kg) | 133 | 176 | 272 | 228 | 229 | 218

the regression equation is \\(\hat{y} = \square + (\square) x\\).
(round the y-intercept to the nearest integer as needed. round the slope to one decimal place as needed.)

Explanation:

Step1: Calculate necessary sums

First, we need to calculate the sums of \( x \), \( y \), \( xy \), and \( x^2 \) for the given data.
Let \( x \) be the overhead width (cm) and \( y \) be the weight (kg).
The data points are:
\( (x_1, y_1) = (7.1, 133) \)
\( (x_2, y_2) = (7.4, 176) \)
\( (x_3, y_3) = (9.8, 272) \)
\( (x_4, y_4) = (9.4, 228) \)
\( (x_5, y_5) = (8.9, 229) \)
\( (x_6, y_6) = (8.5, 218) \)

Calculate \( \sum x \):
\( \sum x = 7.1 + 7.4 + 9.8 + 9.4 + 8.9 + 8.5 = 51.1 \)

Calculate \( \sum y \):
\( \sum y = 133 + 176 + 272 + 228 + 229 + 218 = 1256 \)

Calculate \( \sum xy \):
\( 7.1\times133 + 7.4\times176 + 9.8\times272 + 9.4\times228 + 8.9\times229 + 8.5\times218 \)
\( = 944.3 + 1294.4 + 2665.6 + 2143.2 + 2038.1 + 1853 = 10938.6 \)

Calculate \( \sum x^2 \):
\( 7.1^2 + 7.4^2 + 9.8^2 + 9.4^2 + 8.9^2 + 8.5^2 \)
\( = 50.41 + 54.76 + 96.04 + 88.36 + 79.21 + 72.25 = 440.99 \)

Step2: Calculate slope (\( b \)) and y-intercept (\( a \))

The formula for the slope \( b \) of the regression line is:
\( b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)
where \( n = 6 \) (number of data points).

Substitute the values:
\( n\sum xy = 6\times10938.6 = 65631.6 \)
\( \sum x \sum y = 51.1\times1256 = 64181.6 \)
\( n\sum x^2 = 6\times440.99 = 2645.94 \)
\( (\sum x)^2 = 51.1^2 = 2611.21 \)

Calculate numerator: \( 65631.6 - 64181.6 = 1450 \)
Calculate denominator: \( 2645.94 - 2611.21 = 34.73 \)
So, \( b = \frac{1450}{34.73} \approx 41.8 \) (rounded to one decimal place)

The formula for the y-intercept \( a \) is:
\( a = \frac{\sum y - b\sum x}{n} \)

Substitute the values:
\( b\sum x = 41.8\times51.1 \approx 2135.98 \)
\( \sum y - b\sum x = 1256 - 2135.98 = -879.98 \approx -880 \) (rounded to nearest integer)
\( a = \frac{-880}{6} \approx -146.67 \)? Wait, no, wait, I made a mistake here. Wait, let's recalculate \( a \):

Wait, \( a = \bar{y} - b\bar{x} \), where \( \bar{x} = \frac{\sum x}{n} = \frac{51.1}{6} \approx 8.5167 \), \( \bar{y} = \frac{\sum y}{n} = \frac{1256}{6} \approx 209.3333 \)

So \( a = 209.3333 - 41.8\times8.5167 \)
\( 41.8\times8.5167 \approx 41.8\times8.5 = 355.3, 41.8\times0.0167\approx0.698 \), total \( \approx 355.3 + 0.698 = 355.998 \)
\( a = 209.3333 - 355.998 \approx -146.6647 \approx -147 \)? Wait, but earlier calculation with the formula \( a = \frac{\sum y - b\sum x}{n} \):

\( \sum y - b\sum x = 1256 - 41.8\times51.1 = 1256 - 2135.98 = -879.98 \)
\( a = \frac{-879.98}{6} \approx -146.66 \approx -147 \) (rounded to nearest integer)

Wait, but let's check the slope calculation again. Wait, maybe I made a mistake in the slope. Let's recalculate the slope:

Wait, \( n\sum xy = 6*10938.6 = 65631.6 \)
\( \sum x \sum y = 51.1*1256 = 51.1*1200 + 51.1*56 = 61320 + 2861.6 = 64181.6 \)
Numerator: 65631.6 - 64181.6 = 1450. Correct.
Denominator: \( n\sum x^2 - (\sum x)^2 = 6*440.99 - 51.1^2 = 2645.94 - 2611.21 = 34.73 \). Correct.
So \( b = 1450 / 34.73 ≈ 41.75 ≈ 41.8 \). Correct.

Then \( a = \bar{y} - b\bar{x} = (1256/6) - 41.8*(51.1/6) \)
\( = (1256 - 41.8*51.1)/6 \)
\( 41.8*51.1 = 41.8*(50 + 1.1) = 41.8*50 + 41.8*1.1 = 2090 + 45.98 = 2135.98 \)
\( 1256 - 2135.98 = -879.98 \)
\( -879.98 / 6 ≈ -146.66 ≈ -147 \) (rounded to nearest integer)

Wait, but let's check with another approach. Let's use a calculator for regression.

Alternatively, maybe I made a mistake in the data. Wait, the x values are 7.1,7.4,9.8,9.4,8.9,8.5. Let's recalculate \( \sum x \): 7.1+7.4=14.5; 14.5+9.8=24.3; 24.3+9.4=33.7; 33.7+8.9=42.6; 42.6+8.5=51.1. Correct.

\( \sum y \): 133+176=309; 309+272=581; 581+228=809; 809+229=1038; 1038+218=1256. Correct.

\( \sum xy \): 7.1*133=944.3; 7.4*176=1294.4; 9.8*272=2665.6; 9.4*228=2143.2; 8.9*229=2038.1; 8.5*218=1853. Sum: 944.3+1294.4=2238.7; +2665.6=4904.3; +2143.2=7047.5; +2038.1=9085.6; +1853=10938.6. Correct.

\( \sum x^2 \): 7.1²=50.41; 7.4²=54.76; 9.8²=96.04; 9.4²=88.36; 8.9²=79.21; 8.5²=72.25. Sum: 50.41+54.76=105.17; +96.04=201.21; +88.36=289.57; +79.21=368.78; +72.25=441.03. Wait, earlier I had 440.99, which is due to rounding. So \( \sum x^2 = 441.03 \) (more accurately).

Then denominator: \( n\sum x^2 - (\sum x)^2 = 6*441.03 - 51.1² = 2646.18 - 2611.21 = 34.97 \)

Numerator: 6*10938.6 - 51.1*1256 = 65631.6 - 64181.6 = 1450

So \( b = 1450 / 34.97 ≈ 41.46 ≈ 41.5 \)? Wait, now I'm confused. Wait, maybe the initial slope calculation was wrong. Let's use a calculator for linear regression.

Using a calculator, let's input the data:

x: 7.1,7.4,9.8,9.4,8.9,8.5

y: 133,176,272,228,229,218

Calculating the regression line:

The slope \( b \) is calculated as:

\( b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)

First, calculate \( \bar{x} = 51.1/6 ≈ 8.5167 \), \( \bar{y} = 1256/6 ≈ 209.3333 \)

Calculate \( (x_i - \bar{x}) \) and \( (y_i - \bar{y}) \) for each data point:

1. (7.1 - 8.5167) = -1.4167, (133 - 209.3333) = -76.3333, product: (-1.4167)*(-76.3333) ≈ 108.1, square: (-1.4167)² ≈ 2.007

2. (7.4 - 8.5167) = -1.1167, (176 - 209.3333) = -33.3333, product: (-1.1167)*(-33.3333) ≈ 37.22, square: (-1.1167)² ≈ 1.247

3. (9.8 - 8.5167) = 1.2833, (272 - 209.3333) = 62.6667, product: 1.2833*62.6667 ≈ 80.44, square: (1.2833)² ≈ 1.647

4. (9.4 - 8.5167) = 0.8833, (228 - 209.3333) = 18.6667, product: 0.8833*18.6667 ≈ 16.5, square: (0.8833)² ≈ 0.780

5. (8.9 - 8.5167) = 0.3833, (229 - 209.3333) = 19.6667, product: 0.3833*19.6667 ≈ 7.54, square: (0.3833)² ≈ 0.147

6. (8.5 - 8.5167) = -0.0167, (218 - 209.3333) = 8.6667, product: (-0.0167)*8.6667 ≈ -0.145, square: (-0.0167)² ≈ 0.000278

Now, sum of products (numerator for slope): 108.1 + 37.22 + 80.44 + 16.5 + 7.54 - 0.145 ≈ 249.655

Sum of squares (denominator for slope): 2.007 + 1.247 + 1.647 + 0.780 + 0.147 + 0.000278 ≈ 5.828

Wait, this is different from before. Clearly, my initial method had an error. Wait, no, the formula \( b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \) is correct. Let's recalculate with \( \sum x^2 = 441.03 \) (more accurately):

\( n\sum xy = 6*10938.6 = 65631.6 \)

\( \sum x \sum y = 51.1*1256 = 51.1*1200 + 51.1*56 = 61320 + 2861.6 = 64181.6 \)

\( n\sum x^2 = 6*441.03 = 2646.18 \)

\( (\sum x)^2 = 51.1^2 = 2611.21 \)

So numerator: 65631.6 - 64181.6 = 1450

Denominator: 2646.18 - 2611.21 = 34.97

So \( b = 1450 / 34.97 ≈ 41.46 ≈ 41.5 \) (rounded to one decimal place)

Now, calculate \( a = \bar{y} - b\bar{x} \)

\( \bar{x} = 51.1/6 ≈ 8.5167 \)

\( \bar{y} = 1256/6 ≈ 209.3333 \)

\( a = 209.3333 - 41.5*8.5167 \)

\( 41.5*8.5167 ≈ 41.5*8 + 41.5*0.5167 ≈ 332 + 21.44 ≈ 353.44 \)

\( a = 209.3333 - 353.44 ≈ -144.1067 ≈ -144 \) (rounded to nearest integer)

Answer:

Step1: Calculate necessary sums

First, we need to calculate the sums of \( x \), \( y \), \( xy \), and \( x^2 \) for the given data.
Let \( x \) be the overhead width (cm) and \( y \) be the weight (kg).
The data points are:
\( (x_1, y_1) = (7.1, 133) \)
\( (x_2, y_2) = (7.4, 176) \)
\( (x_3, y_3) = (9.8, 272) \)
\( (x_4, y_4) = (9.4, 228) \)
\( (x_5, y_5) = (8.9, 229) \)
\( (x_6, y_6) = (8.5, 218) \)

Calculate \( \sum x \):
\( \sum x = 7.1 + 7.4 + 9.8 + 9.4 + 8.9 + 8.5 = 51.1 \)

Calculate \( \sum y \):
\( \sum y = 133 + 176 + 272 + 228 + 229 + 218 = 1256 \)

Calculate \( \sum xy \):
\( 7.1\times133 + 7.4\times176 + 9.8\times272 + 9.4\times228 + 8.9\times229 + 8.5\times218 \)
\( = 944.3 + 1294.4 + 2665.6 + 2143.2 + 2038.1 + 1853 = 10938.6 \)

Calculate \( \sum x^2 \):
\( 7.1^2 + 7.4^2 + 9.8^2 + 9.4^2 + 8.9^2 + 8.5^2 \)
\( = 50.41 + 54.76 + 96.04 + 88.36 + 79.21 + 72.25 = 440.99 \)

Step2: Calculate slope (\( b \)) and y-intercept (\( a \))

The formula for the slope \( b \) of the regression line is:
\( b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)
where \( n = 6 \) (number of data points).

Substitute the values:
\( n\sum xy = 6\times10938.6 = 65631.6 \)
\( \sum x \sum y = 51.1\times1256 = 64181.6 \)
\( n\sum x^2 = 6\times440.99 = 2645.94 \)
\( (\sum x)^2 = 51.1^2 = 2611.21 \)

Calculate numerator: \( 65631.6 - 64181.6 = 1450 \)
Calculate denominator: \( 2645.94 - 2611.21 = 34.73 \)
So, \( b = \frac{1450}{34.73} \approx 41.8 \) (rounded to one decimal place)

The formula for the y-intercept \( a \) is:
\( a = \frac{\sum y - b\sum x}{n} \)

Substitute the values:
\( b\sum x = 41.8\times51.1 \approx 2135.98 \)
\( \sum y - b\sum x = 1256 - 2135.98 = -879.98 \approx -880 \) (rounded to nearest integer)
\( a = \frac{-880}{6} \approx -146.67 \)? Wait, no, wait, I made a mistake here. Wait, let's recalculate \( a \):

Wait, \( a = \bar{y} - b\bar{x} \), where \( \bar{x} = \frac{\sum x}{n} = \frac{51.1}{6} \approx 8.5167 \), \( \bar{y} = \frac{\sum y}{n} = \frac{1256}{6} \approx 209.3333 \)

So \( a = 209.3333 - 41.8\times8.5167 \)
\( 41.8\times8.5167 \approx 41.8\times8.5 = 355.3, 41.8\times0.0167\approx0.698 \), total \( \approx 355.3 + 0.698 = 355.998 \)
\( a = 209.3333 - 355.998 \approx -146.6647 \approx -147 \)? Wait, but earlier calculation with the formula \( a = \frac{\sum y - b\sum x}{n} \):

\( \sum y - b\sum x = 1256 - 41.8\times51.1 = 1256 - 2135.98 = -879.98 \)
\( a = \frac{-879.98}{6} \approx -146.66 \approx -147 \) (rounded to nearest integer)

Wait, but let's check the slope calculation again. Wait, maybe I made a mistake in the slope. Let's recalculate the slope:

Wait, \( n\sum xy = 6*10938.6 = 65631.6 \)
\( \sum x \sum y = 51.1*1256 = 51.1*1200 + 51.1*56 = 61320 + 2861.6 = 64181.6 \)
Numerator: 65631.6 - 64181.6 = 1450. Correct.
Denominator: \( n\sum x^2 - (\sum x)^2 = 6*440.99 - 51.1^2 = 2645.94 - 2611.21 = 34.73 \). Correct.
So \( b = 1450 / 34.73 ≈ 41.75 ≈ 41.8 \). Correct.

Then \( a = \bar{y} - b\bar{x} = (1256/6) - 41.8*(51.1/6) \)
\( = (1256 - 41.8*51.1)/6 \)
\( 41.851.1 = 41.8(50 + 1.1) = 41.850 + 41.81.1 = 2090 + 45.98 = 2135.98 \)
\( 1256 - 2135.98 = -879.98 \)
\( -879.98 / 6 ≈ -146.66 ≈ -147 \) (rounded to nearest integer)

Wait, but let's check with another approach. Let's use a calculator for regression.

Alternatively, maybe I made a mistake in the data. Wait, the x values are 7.1,7.4,9.8,9.4,8.9,8.5. Let's recalculate \( \sum x \): 7.1+7.4=14.5; 14.5+9.8=24.3; 24.3+9.4=33.7; 33.7+8.9=42.6; 42.6+8.5=51.1. Correct.

\( \sum y \): 133+176=309; 309+272=581; 581+228=809; 809+229=1038; 1038+218=1256. Correct.

\( \sum xy \): 7.1133=944.3; 7.4176=1294.4; 9.8272=2665.6; 9.4228=2143.2; 8.9229=2038.1; 8.5218=1853. Sum: 944.3+1294.4=2238.7; +2665.6=4904.3; +2143.2=7047.5; +2038.1=9085.6; +1853=10938.6. Correct.

\( \sum x^2 \): 7.1²=50.41; 7.4²=54.76; 9.8²=96.04; 9.4²=88.36; 8.9²=79.21; 8.5²=72.25. Sum: 50.41+54.76=105.17; +96.04=201.21; +88.36=289.57; +79.21=368.78; +72.25=441.03. Wait, earlier I had 440.99, which is due to rounding. So \( \sum x^2 = 441.03 \) (more accurately).

Then denominator: \( n\sum x^2 - (\sum x)^2 = 6*441.03 - 51.1² = 2646.18 - 2611.21 = 34.97 \)

Numerator: 610938.6 - 51.11256 = 65631.6 - 64181.6 = 1450

So \( b = 1450 / 34.97 ≈ 41.46 ≈ 41.5 \)? Wait, now I'm confused. Wait, maybe the initial slope calculation was wrong. Let's use a calculator for linear regression.

Using a calculator, let's input the data:

x: 7.1,7.4,9.8,9.4,8.9,8.5

y: 133,176,272,228,229,218

Calculating the regression line:

The slope \( b \) is calculated as:

\( b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)

First, calculate \( \bar{x} = 51.1/6 ≈ 8.5167 \), \( \bar{y} = 1256/6 ≈ 209.3333 \)

Calculate \( (x_i - \bar{x}) \) and \( (y_i - \bar{y}) \) for each data point:

1. (7.1 - 8.5167) = -1.4167, (133 - 209.3333) = -76.3333, product: (-1.4167)*(-76.3333) ≈ 108.1, square: (-1.4167)² ≈ 2.007

1. (7.4 - 8.5167) = -1.1167, (176 - 209.3333) = -33.3333, product: (-1.1167)*(-33.3333) ≈ 37.22, square: (-1.1167)² ≈ 1.247

1. (9.8 - 8.5167) = 1.2833, (272 - 209.3333) = 62.6667, product: 1.2833*62.6667 ≈ 80.44, square: (1.2833)² ≈ 1.647

1. (9.4 - 8.5167) = 0.8833, (228 - 209.3333) = 18.6667, product: 0.8833*18.6667 ≈ 16.5, square: (0.8833)² ≈ 0.780

1. (8.9 - 8.5167) = 0.3833, (229 - 209.3333) = 19.6667, product: 0.3833*19.6667 ≈ 7.54, square: (0.3833)² ≈ 0.147

1. (8.5 - 8.5167) = -0.0167, (218 - 209.3333) = 8.6667, product: (-0.0167)*8.6667 ≈ -0.145, square: (-0.0167)² ≈ 0.000278

Now, sum of products (numerator for slope): 108.1 + 37.22 + 80.44 + 16.5 + 7.54 - 0.145 ≈ 249.655

Sum of squares (denominator for slope): 2.007 + 1.247 + 1.647 + 0.780 + 0.147 + 0.000278 ≈ 5.828

Wait, this is different from before. Clearly, my initial method had an error. Wait, no, the formula \( b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \) is correct. Let's recalculate with \( \sum x^2 = 441.03 \) (more accurately):

\( n\sum xy = 6*10938.6 = 65631.6 \)

\( \sum x \sum y = 51.1*1256 = 51.1*1200 + 51.1*56 = 61320 + 2861.6 = 64181.6 \)

\( n\sum x^2 = 6*441.03 = 2646.18 \)

\( (\sum x)^2 = 51.1^2 = 2611.21 \)

So numerator: 65631.6 - 64181.6 = 1450

Denominator: 2646.18 - 2611.21 = 34.97

So \( b = 1450 / 34.97 ≈ 41.46 ≈ 41.5 \) (rounded to one decimal place)

Now, calculate \( a = \bar{y} - b\bar{x} \)

\( \bar{x} = 51.1/6 ≈ 8.5167 \)

\( \bar{y} = 1256/6 ≈ 209.3333 \)

\( a = 209.3333 - 41.5*8.5167 \)

\( 41.5*8.5167 ≈ 41.5*8 + 41.5*0.5167 ≈ 332 + 21.44 ≈ 353.44 \)

\( a = 209.3333 - 353.44 ≈ -144.1067 ≈ -144 \) (rounded to nearest integer)