A manufacturer stores drums of chemicals - Edexcel - A-Level Maths Statistics - Question 3 - 2006 - Paper 1
Question 3
A manufacturer stores drums of chemicals. During storage, evaporation takes place. A random sample of 10 drums was taken and the time in storage, x weeks, and the ev... show full transcript
Worked Solution & Example Answer:A manufacturer stores drums of chemicals - Edexcel - A-Level Maths Statistics - Question 3 - 2006 - Paper 1
Step 1
Draw a scatter diagram
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Answer
To represent the given data, plot the points on a graph with 'Time (weeks)' on the x-axis and 'Evaporation loss (ml)' on the y-axis. Each point corresponds to the (x, y) values from the table. Ensure to label the axes and provide a title for clarity.
Step 2
Give a reason to support fitting a regression model
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Answer
The points in the scatter plot lie close to a straight line, indicating a linear relationship between the time in storage and evaporation loss. This justifies the use of a linear regression model to describe the data.
Step 3
Find, to 2 decimal places, the value of a and the value of b
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Using the provided formulas:
First, calculate the slope (b):
Sy=8354,Sx=1352,Syy=53112,
Plugging the values into the formula:
b=228.4891.6≈3.90
Then calculate the intercept (a) using:
a=yˉ−bxˉ=29.02−3.90⋅7.04≈29
Thus, b ≈ 3.90 and a ≈ 29.00.
Step 4
Give an interpretation of the value of b
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The value of b indicates the rate of evaporation. Specifically, for every additional week in storage, approximately 3.90 ml of chemical evaporates, reflecting a consistent trend over time.
Step 5
Using your model, predict the amount of evaporation after 19 weeks
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Using the linear regression model:
y=a+b⋅x=29+3.90⋅19=29+74.1=103.10ml
Step 6
Using your model, predict the amount of evaporation after 35 weeks
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Applying the same model for 35 weeks:
y=29+3.90⋅35=29+136.5=165.52ml
Step 7
Comment on the reliability of each of your predictions
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The prediction for 19 weeks is fairly reliable as it falls within the observed range of x values. However, the prediction for 35 weeks is less reliable, as it exceeds the maximum observed time, suggesting that the linear model may not hold true for such extrapolations.
