Photo AI
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
-
No Scale on the Y-axis: The graph does not start at zero, which can exaggerate the visual impact of any changes. This means even small increases in salt consumption might appear as significant declines if the scale is misleading.
-
Consumption Data: Gemma may refer to the distinction between salt purchased and actual consumption. People might have purchased less salt but still used similar amounts, reflecting usage trends that do not correlate directly with purchase data.
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
-
Formulate Hypotheses:
- Null Hypothesis ( H_0): μ = 78.9 grams (the mean sugar purchase has not changed).
- Alternative Hypothesis ( H_a): μ ≠ 78.9 grams (the mean sugar purchase has changed).
-
Calculate the Test Statistic:
The formula is:
z = rac{ar{x} - ext{μ}}{ rac{ ext{σ}}{ ext{√n}}}Substituting values:
z = rac{80.4 - 78.9}{ rac{25.0}{ ext{√918}}} \ \ = rac{1.5}{0.8165} \ \ ext{(approx.)} = 1.83 -
Determine Critical Value and Compare:
- For a two-tailed test at the 5% significance level, the critical values are -1.96 and +1.96.
- Since 1.83 is less than 1.96, we fail to reject the null hypothesis.
-
Conclusion: There is insufficient evidence at the 5% level of significance to suggest that the mean amount of sugar purchased per person has changed.
Step 3
With reference to the 10% significance level, explain why it is not necessarily true that there has been a change.
Answer
-
Type I Error: At the 10% significance level, there is a 10% chance of rejecting the null hypothesis when it is actually true. Therefore, even though we rejected the null hypothesis, there is still a likelihood that this conclusion could be incorrect.
-
Caution in Interpretation: The rejection indicates some evidence for change, but it does not confirm a definite change. Statistical significance does not equate to practical significance, and further confirmation would be necessary to assert a real change in consumption.