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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, $E_w$, and the average weekly temperature, $T_°C$, for 8 weeks during the su... 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 correlation between sales figures and average weekly temperature, we can set up our hypotheses as follows:
- Null hypothesis (): There is no negative correlation between sales figures and average weekly temperature ().
- Alternative hypothesis (): There is a negative correlation between sales figures and average weekly temperature ().
Using the provided correlation coefficient of , we can calculate the test statistic:
t = rac{r ext{ } imes ext{ } ext{ sqrt}(n-2)}{ ext{ sqrt}(1 - r^2)}
where is the number of pairs of observations (in this case, ).
Now substituting the values: t = rac{-0.915 imes ext{sqrt}(8 - 2)}{ ext{sqrt}(1-(-0.915)^2)}
Calculating the value, we can compare it to the critical value from the t-distribution table at 5% significance level with degrees of freedom to make our decision on .
Step 2
Suggest a possible reason for this correlation.
Answer
A possible reason for the negative correlation between sales figures and average weekly temperature could be that customers prefer to shop for clothes when the weather is cooler. During warmer weeks, people may choose outdoor activities over shopping, thereby reducing sales.
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 indicates a strong negative relationship between sales figures and average weekly temperature, which is consistent with Tessa's suggestion of using a linear regression model. A strong negative correlation supports the idea that as temperature rises, sales tend to decrease, justifying the use of linear regression to model this relationship.
Step 4
Step 5
Give an interpretation of the gradient of this regression equation.
Answer
In Tessa's regression equation, , the gradient (or slope) of -171 indicates that for each one-degree increase in average weekly temperature, the weekly sales figures are expected to decrease by approximately $171. This negative gradient quantifies the expected decline in sales as temperatures rise.