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The graph below shows the amount of salt, in grams, purchased per person per week in England between 2001–02 and 2014, based upon the Large Data Set - AQA - A-Level Maths: Pure - Question 16 - 2019 - Paper 3
Question 16
The graph below shows the amount of salt, in grams, purchased per person per week in England between 2001–02 and 2014, based upon the Large Data Set. Meera and Gemm... show full transcript
Worked Solution & Example Answer:The graph below shows the amount of salt, in grams, purchased per person per week in England between 2001–02 and 2014, based upon the Large Data Set - AQA - A-Level Maths: Pure - Question 16 - 2019 - Paper 3
Step 1
Give two reasons why Gemma may be correct.
Answer
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No Scale on the Y-Axis: The graph does not start at zero, making it appear that there was a greater fluctuation in salt consumption than there actually was when viewed on a different scale. This creates a misleading impression about the level of salt consumed.
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Comparison with Other Data: Gemma could argue that the salt purchased may have been categorized differently over the years, such as separating processed and fresh foods which can lead to an underestimation of total consumption. The data might not provide a valid comparison due to this change in data categorization.
Step 2
Investigate, at the 5% level of significance, whether the mean amount of sugar purchased per person in England has changed between 2014 and 2018.
Answer
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State the Hypotheses:
- Null hypothesis (): ar{ ext{x}} = 78.9 grams (mean amount of sugar has not changed)
- Alternative hypothesis (): ar{ ext{x}} eq 78.9 grams (mean amount of sugar has changed)
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Calculate the Test Statistic:
- Using the formula: ext{Test statistic} = rac{ar{X} - ext{population mean}}{ rac{ ext{standard deviation}}{ ext{sqrt(n)}}}
- For our case: ext{Test statistic} = rac{80.4 - 78.9}{ rac{25.0}{ ext{sqrt}(918)}} ext{ (compute this)}
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Obtain the Critical Value:
- For a two-tailed test at and , the critical value is approximately .
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Decision Rule:
- If the test statistic falls outside the interval (), we reject the null hypothesis.
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Comparison and Conclusion:
- Compare the test statistic to the critical value. If it falls outside the critical value, conclude that there is sufficient evidence to suggest a change in mean sugar consumption.
Step 3
With reference to the 10% significance level, explain why it is not necessarily true that there has been a change.
Answer
At the 10% significance level, rejecting the null hypothesis only indicates that there is a 10% chance of making a Type I error, meaning that the null hypothesis is actually true when we have rejected it.
Thus, while the hypothesis has been rejected, it does not conclusively prove that a change has occurred. It merely suggests that the data collected provides sufficient evidence to lean towards a change, but other factors could influence this outcome. Therefore, we cannot definitively state that a change has happened based solely on the statistical test results.