Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, $w$, and the average weekly temperature, $t^{ ext{C}}$, for 8 weeks during the summer. The product moment correlation coefficient for these data is $r = -0.915$. (a) 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. (b) Suggest a possible reason for this correlation. Tessa suggests that a linear regression model could be used to model these data. (c) State, giving a reason, whether or not the correlation coefficient is consistent with Tessa’s suggestion. (d) State, giving a reason, which variable would be the explanatory variable. Tessa calculated the linear regression equation as $w = 10 755 - 171t$ (e) Give an interpretation of the gradient of this regression equation.

A-Level Edexcel Maths Mechanics Energy: Tessa owns a small clothes shop

Tessa owns a small clothes shop in a seaside town - Edexcel - A-Level Maths Mechanics - Question 2 - 2018 - Paper 1

Question 2

Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, $w$, and the average weekly temperature, $t^{ ext{C}}$, for 8 weeks during 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 1

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.

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Answer

The null hypothesis ( $H_0$ ) states that there is no correlation between the sales figures and average weekly temperature, hence $ho = 0$ . The alternative hypothesis ( $H_1$ ) states that there is a negative correlation, hence $ho < 0$ . Given the significance level of 5%, we compare the computed correlation coefficient ( $r = -0.915$ ) with the critical value of $-0.6215$ . Since $-0.915 < -0.6215$ , we reject the null hypothesis. There is sufficient evidence to suggest a negative correlation between sales figures and temperature.

Step 2

Suggest a possible reason for this correlation.

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Answer

A potential reason for this correlation is that as temperatures increase, people tend to spend more time outdoors, particularly at the beach, instead of shopping in stores. Therefore, it's reasonable to conclude that higher temperatures may lead to lower sales figures.

Step 3

State, giving a reason, whether or not the correlation coefficient is consistent with Tessa’s suggestion.

96%

101 rated

Answer

The correlation coefficient of $r = -0.915$ is indeed consistent with Tessa's suggestion of using a linear regression model, as it indicates a strong negative correlation between the variables.

Step 4

State, giving a reason, which variable would be the explanatory variable.

98%

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Answer

The explanatory variable would be the average weekly temperature ( $t$ ) since the sales figures ( $w$ ) are likely to depend on the temperature; as $t$ changes, it has a direct influence on the sales.

Step 5

Give an interpretation of the gradient of this regression equation.

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117 rated

Answer

The gradient of the regression equation $w = 10 755 - 171t$ indicates that for every degree rise in temperature, there is a decrease of £171 in the weekly sales figures. This suggests a detrimental effect of increasing temperatures on sales.

Tessa owns a small clothes shop in a seaside town - Edexcel - A-Level Maths Mechanics - Question 2 - 2018 - Paper 1

Worked Solution & Example Answer:Tessa owns a small clothes shop in a seaside town - Edexcel - A-Level Maths Mechanics - Question 2 - 2018 - Paper 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.

Suggest a possible reason for this correlation.

State, giving a reason, whether or not the correlation coefficient is consistent with Tessa’s suggestion.

State, giving a reason, which variable would be the explanatory variable.

Give an interpretation of the gradient of this regression equation.

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