Statistics · Guided Practice
Linear Regression
9 questions with answers and video solutions. Try each one before revealing the answer.
Interpret the value in the context of the study.
Answer
Use the model to predict sales on a day with hours of sunshine.
Answer
About ice creams.
Interpret the value , and comment on whether it is meaningful.
Answer
It is only meaningful if days with no sunshine were included in the data used to build the model; otherwise it is an extrapolation beyond the data.
Use the model to predict the calories burned on a km run.
Answer
About calories.
Explain why the intercept of cannot be interpreted sensibly.
Answer
The model was built from runs of a positive length, so a distance of lies outside the data it applies to.
Explain whether the model should be used to predict the efficiency of a kg car.
Answer
kg lies outside the range of the data used to build the model, so the prediction is an extrapolation and there is no evidence the relationship continues that far.
Answer
Interpolation is more reliable, because the model has been fitted to data covering that range.
You must refer to the information on 'Strength and conditioning' given in the pre-release material when answering this question.
You must also refer to the spreadsheet file 'Q7 Jump.csv' for the data, and the word processing file 'Q7 Jump Answers.docx' when answering this question.
You must complete parts (a) (i), (b) (i), (b) (ii), and (c) using appropriate statistical software.
You must include all output from statistical software, and your answers in the word processing file 'Q7 Jump Answers.docx'.
A strength and conditioning coach wants to increase vertical jump height performance in their trainees. The data in the spreadsheet file shows back squat weight (kg) and vertical jump height (cm).
(a)(i) Construct a scatter plot of vertical jump height on back squat weight for the data.
(a)(ii) Make an appropriate comment about the relationship between vertical jump height and back squat weight.
(b)(i) Find the correlation coefficient between back squat weight and vertical jump height.
(b)(ii) Find the equation of the regression line of vertical jump height on back squat weight.
(c) Use your statistical software to estimate the vertical jump height for a trainee who can back squat 165 kg, and comment on the accuracy of the predicted value.
Based on the correlation, the coach advises the trainees that increasing their back squat weight will increase their vertical jump height.
(d) Explain why the statistical analysis does not support this advice.
Answer
(a)(i) Generate scatterplot from software with appropriate title and axis labels.
(a)(ii) Approximately a positive linear relationship (or as back squat weight increases, vertical jump height increases).
(b)(i)
(b)(ii) jump height = back squat +
(c) cm. Since kg is within the range of the data used to make the model and the model is a strong linear model, the prediction is fairly accurate.
(d) Correlation is not causation.
You must refer to the spreadsheet file 'Q8 Biomass Data' when answering this question.
You must complete parts (a) (i), (b) and (c) using statistical software.
You must copy and paste your answers to parts (a) (i), (b) and (c) into the word processing file 'Q8 Biomass Answers'.
The UK has a varied mix of renewable technologies and fuels including biomass which is a key fuel source for the decarbonisation of electricity generation and heat provision. Woodchips are an example of a source of biomass. The heat output of woodchips used to generate energy varies depending on moisture content. The data in the spreadsheet file shows moisture content (%) and the associated heat outputs (kilowatts) of various random samples of woodchip.
(a) (i) Construct a scatter diagram for the data.
(ii) Make two comments about the scatter diagram.
(b) Find the equation of the regression line of heat output on percentage moisture content.
(c) Estimate the heat output of woodchips with a moisture content of 35% and interpret this estimate by referring to a prediction interval.
(d) Explain the implication of your analysis for anyone intending to use woodchips as a source of heat.
Answer
(a)(i) Scatterplot constructed with heat output (kW) on the y-axis and moisture content (%) on the x-axis.
(a)(ii) e.g., There is a linear relationship. There is a strong negative association.
(b) heat output =
(c) The estimated heat output of woodchip with a moisture content of 35% is 5.9 kW, however the true value is likely to be between 5.3 and 6.6 kW.
(d) The lower the percentage moisture content of the woodchip, the greater the heat output.