A-Level Edexcel Maths Statistics Hypothesis Testing: Barbara is investigating the rel

Barbara is investigating the relationship between average income (GDP per capita), $x$ US dollars, and average annual carbon dioxide (CO₂) emissions, $y$ tonnes, for different countries - Edexcel - A-Level Maths Statistics - Question 3 - 2019 - Paper 1

Question 3

Barbara is investigating the relationship between average income (GDP per capita), $x$ US dollars, and average annual carbon dioxide (CO₂) emissions, $y$ tonnes, for... show full transcript

Worked Solution & Example Answer:Barbara is investigating the relationship between average income (GDP per capita), $x$ US dollars, and average annual carbon dioxide (CO₂) emissions, $y$ tonnes, for different countries - Edexcel - A-Level Maths Statistics - Question 3 - 2019 - Paper 1

Step 1

Stating your hypotheses clearly, test, at the 5% level of significance, whether or not the product moment correlation coefficient for all countries is greater than zero.

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Answer

To test the hypothesis, we first set our null and alternative hypotheses:

Null Hypothesis ( $H_0$ ): $ho = 0$ (there is no correlation)
Alternative Hypothesis ( $H_1$ ): $ho > 0$ (there is a positive correlation)

The significance level is set at $eta = 0.05$ . To perform the test, we need the critical value for the correlation coefficient. The critical value for 24 samples at a 5% significance level (one-tailed) is approximately 0.3438.

Since our calculated product moment correlation coefficient is 0.446, we will compare this to our critical value:

Since 0.446 > 0.3438, we reject the null hypothesis.

Thus, we conclude that there is sufficient evidence to suggest that the product moment correlation coefficient for all countries is greater than zero.

Step 2

Explain how this value supports Barbara’s belief.

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Answer

The product moment correlation coefficient of 0.882 between $e$ and $m$ indicates a strong positive linear relationship. This supports Barbara's belief that a non-linear model may be more appropriate. Since the correlation value is significantly higher than what was originally observed (0.446), it suggests that the transformation of the data using logarithmic coding ( $m = ext{log}_10 x$ , $c = ext{log}_10 y$ ) indeed enhances the relationship, implying that the original relationship may not be best represented by a linear model.

Step 3

Show that the relationship between $y$ and $x$ can be written in the form $y = ax^r$ where $a$ and $r$ are constants to be found.

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Answer

Starting from the logarithmic model obtained by coding the data:

$ext{log}_{10} y = -1.82 + 0.89 ext{log}_{10} x$

We convert this back to exponential form:

Rewrite it as: $ext{log}_{10} y = ext{log}_{10}(10^{-1.82}) + 0.89 ext{log}_{10} x$
This implies: $y = 10^{-1.82} imes x^{0.89}$

Now, defining constants:

Let $a = 10^{-1.82}$ and $r = 0.89$ , we have: $y = ax^r$ Thus, we have shown the relationship is in the required form with $a$ and $r$ found.

Barbara is investigating the relationship between average income (GDP per capita), $x$ US dollars, and average annual carbon dioxide (CO₂) emissions, $y$ tonnes, for different countries - Edexcel - A-Level Maths Statistics - Question 3 - 2019 - Paper 1

Worked Solution & Example Answer:Barbara is investigating the relationship between average income (GDP per capita), $x$ US dollars, and average annual carbon dioxide (CO₂) emissions, $y$ tonnes, for different countries - Edexcel - A-Level Maths Statistics - Question 3 - 2019 - Paper 1

Stating your hypotheses clearly, test, at the 5% level of significance, whether or not the product moment correlation coefficient for all countries is greater than zero.

Explain how this value supports Barbara’s belief.

Show that the relationship between $y$ and $x$ can be written in the form $y = ax^r$ where $a$ and $r$ are constants to be found.

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