5.09e Use regression: for estimation in context

A baker is aware that the pH of his sourdough, \(y\), and the hydration, \(x\), affect the taste and texture of the final product. The hydration is measured in ml of water per 100 g of flour (ml/100 g). The baker researches how the pH of his sourdough changes as the hydration changes. The results of his research are shown in the diagram below. \includegraphics{figure_5}

1. Describe the relationship between pH and hydration. [2]
2. The equation of the regression line for \(y\) on \(x\) is $$y = 5.4 - 0.02x.$$
   1. Interpret the gradient and intercept of the regression line in this context.
   2. Estimate the pH of the sourdough when the hydration is 20 ml/100 g. Comment on the reliability of this estimate. [4]

A university professor conducted some research into factors that affect job satisfaction. The four factors considered were Interesting work, Good wages, Job security and Appreciation of work done. The professor interviewed workers at 14 different companies and asked them to rate their companies on each of the factors. The workers' ratings were averaged to give each company a score out of 5 on each factor. Each company was also given a score out of 100 for Job satisfaction. The following graph shows the part of the research concerning Job Satisfaction versus Interesting work. \includegraphics{figure_2}

1. Calculate the equation of the least squares regression line of Job satisfaction (\(y\)) on Interesting work (\(x\)), given the following summary statistics. [5] \(\sum x = 46 \cdot 2\), \quad \(\sum y = 898\), \quad \(S_{xx} = 3 \cdot 48\) \(S_{xy} = 49 \cdot 45\), \quad \(S_{yy} = 1437 \cdot 714\), \quad \(n = 14\)
2. Give two reasons why it would be inappropriate for the professor to use this equation to calculate the score for Interesting work from a Job satisfaction score of 90. [2]

For a set of 30 pairs of observations of the variables \(x\) and \(y\), it is known that \(\sum x = 420\) and \(\sum y = 240\). The least squares regression line of \(y\) on \(x\) passes through the point with coordinates \((19, 20)\).

1. Show that the equation of the regression line of \(y\) on \(x\) is \(y = 2 \cdot 4x - 25 \cdot 6\) and use it to predict the value of \(y\) when \(x = 26\). [6]
2. State two reasons why your prediction in part (a) may not be reliable. [2]

9 For a random sample of 10 observations of pairs of values \(( x , y )\), the equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\) are $$y = 4.21 x - 0.862 \quad \text { and } \quad x = 0.043 y + 6.36 ,$$ respectively.

1. Find the value of the product moment correlation coefficient for the sample.
2. Test, at the \(10 \%\) significance level, whether there is evidence of non-zero correlation between the variables.
3. Find the mean values of \(x\) and \(y\) for this sample.
4. Estimate the value of \(x\) when \(y = 2.3\) and comment on the reliability of your answer.