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Tessa owns a small clothes shop in a seaside town - Edexcel - A-Level Maths Mechanics - Question 2 - 2018 - Paper 2
Question 2
Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, £w, and the average weekly temperature, °C, for 8 weeks during the summer. T... show full transcript
Worked Solution & Example Answer:Tessa owns a small clothes shop in a seaside town - Edexcel - A-Level Maths Mechanics - Question 2 - 2018 - Paper 2
Step 1
Stating your hypotheses clearly and using a 5% level of significance, test whether or not the correlation between sales figures and average weekly temperature is negative.
Answer
To test the hypothesis, we define the null hypothesis (H0) as there being no correlation between sales figures and average weekly temperature (correlation coefficient = 0). The alternative hypothesis (H1) states that there is a negative correlation (correlation coefficient < 0).
Given the product moment correlation coefficient of -0.915, we compare this to critical values from the t-distribution table based on n-2 degrees of freedom (where n is the number of data points, which is 8).
Using a significance level of 5%, the critical value for one-tailed test with 6 degrees of freedom is -2.447. The calculated t-value can be derived from:
where r is the correlation coefficient and n is the sample size. This yields:
Since -4.103 < -2.447, we reject the null hypothesis H0. There is significant evidence at the 5% level to conclude that there is a negative correlation between sales figures and average weekly temperature.
Step 2
Suggest a possible reason for this correlation.
Answer
A possible reason for this correlation could be that colder weeks lead to less foot traffic in Tessa's shop, resulting in lower sales figures. Customers may prefer to stay indoors during cooler weather, which could affect the sales of clothing typically associated with warmer temperatures.
Step 3
State, giving a reason, whether or not the correlation coefficient is consistent with Tessa’s suggestion.
Answer
The correlation coefficient of -0.915 strongly indicates a negative correlation, which is consistent with Tessa's suggestion that a linear regression model could be employed. A strong negative correlation suggests that as the average weekly temperature decreases, the sales figures are likely to decrease as well.
Step 4
Step 5
Give an interpretation of the gradient of this regression equation.
Answer
The gradient of the regression equation w = 10 755 – 171t represents the average change in weekly sales figures (£w) for each one-degree increase in average weekly temperature (°C). Specifically, a gradient of -171 indicates that for every one degree increase in temperature, sales are expected to decrease by £171.