Regression 1

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Ruth Hand

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2 years, 7 months ago

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Christian Lawson-Perfect 2 years, 6 months ago

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Question statement

A manufacturing plant produces plastic pellets. To improve the efficiency of approximate stock checks, the manager wants to be able to estimate how many pellets are left in partially full containers. A random sample of 12 partially full containers is taken and each container is weighed and the corresponding number of pellets counted. The results are shown in the table below:

Weight of container and pellets ( $x$ ) g	Number of counted pellets ( $y$ )
15	80
20	100
20	110
25	130
25	140
25	160
30	170
35	200
35	210
40	210
40	230
40	240

Give any introductory information the student needs.

			Name	Type	Generated Value

sumx	integer	350
sumx2	integer	11050
sumy	integer	1980
sumxy	integer	62800
b	rational	6
a	rational	-10
num1	integer	21
num2	integer	36
num3	integer	49
sol1	rational	116
sol2	rational	206
sol3	rational	284

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Name

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Value

xxxxxxxxxx
 
350

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Generated value: `integer`

350

→ Used by:

Advice

Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill

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Find the least-squares regression line in the form $y = mx+c$ .

$y=$ Unnamed gap

Advice

Part (a)

To calculate the regression line $y=a+bx$ we need to calculate $a$ and $b$ , where

$b = \frac{\frac{\sum{xy}}{n}-\frac{\sum{x}}{n}\frac{\sum{y}}{n}}{\frac{\sum{x^2}}{n}-\left(\frac{\sum{x}}{n}\right)^2} \quad \text{and} \quad a = \bar{y}-b \bar{x}.$

From the table, we find

$\sum{x} = \var{sumx}, \quad \sum{y} = \var{sumy}, \quad \sum{x^2}=\var{sumx2}, \quad \sum{xy} = \var{sumxy}.$

Plugging these values into the formulae for $a$ and $b$ , we find that $a=\var{a}$ and $b=\var{b}$ . Therefore, the least squares regression line is

$y = \simplify{{b}x+{a}}.$

Part (b)

Using the result from Part (a), the predicted number of pellets are

$\begin{split} \text{(i)}\,\,y_{\var{num1}} &\,= \var{b} \times \var{num1} - \var{-a} = \var{sol1}; \\ \text{(ii)}\,\, y_{\var{num2}} &\,= \var{b} \times \var{num2} - \var{-a} = \var{sol2}; \\ \text{(iii)}\,\, y_{\var{num3}} &\,= \var{b} \times \var{num3} - \var{-a} = \var{sol3}. \end{split}$

Note: For the final result, we have assumed that a linear relationship continues beyond the range of values taken.

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This question is used in the following exam:

Correlation and regression by Ruth Hand in MASH Bath: Moodle quizzes and TS.

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Christian Lawson-Perfect 2 years, 6 months ago

Ruth Hand 2 years, 7 months ago

Ruth Hand 3 years, 7 months ago

Ben McGovern 4 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Regression 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 2 years, 6 months ago

Ruth Hand 2 years, 7 months ago

Ruth Hand 3 years, 7 months ago

Ben McGovern 4 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`