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A survey during 2013 investigated mean expenditure on bread and on alcohol - AQA - A-Level Maths Mechanics - Question 14 - 2019 - Paper 3
Question 14
A survey during 2013 investigated mean expenditure on bread and on alcohol. The 2013 survey obtained information from 12 144 adults. The survey revealed that the m... show full transcript
Worked Solution & Example Answer:A survey during 2013 investigated mean expenditure on bread and on alcohol - AQA - A-Level Maths Mechanics - Question 14 - 2019 - Paper 3
Step 1
Carry out a hypothesis test, at the 5% significance level
Answer
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Define the Hypotheses:
- Null Hypothesis (H0): μ = 123
- Alternative Hypothesis (H1): μ ≠ 123
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Find the Test Statistic: Using the formula:
z = rac{ar{x} - ext{μ}}{ rac{σ}{ ext{sqrt}(n)}} = rac{127 - 123}{ rac{70}{ ext{sqrt}(12144)}}This yields a test statistic of 6.30.
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Determine the Critical Value: For a two-tailed test at 5% significance level, the critical values are ±1.96.
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Decision: Since the test statistic 6.30 is greater than 1.96, we reject the null hypothesis, suggesting that there is evidence to indicate the mean expenditure on bread has changed from 2012 to 2013.
Step 2
Calculate the greatest and least values for the sample mean expenditure
Answer
To determine the acceptance range of the null hypothesis, we calculate:
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Greatest value:
ext{Upper Limit} = ext{μ} + z_{ ext{critical}} imes rac{σ}{ ext{sqrt}(n)} = 123 + 1.96 imes rac{70}{ ext{sqrt}(12144)}This gives approximately 127.10.
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Least value:
ext{Lower Limit} = ext{μ} - z_{ ext{critical}} imes rac{σ}{ ext{sqrt}(n)} = 123 - 1.96 imes rac{70}{ ext{sqrt}(12144)}This gives approximately 118.90.
Step 3
State with a reason the test result supports the following statements: 14 (b)(i)
Answer
The statement "the mean UK expenditure on alcohol per adult per week increased by 17p from 2009 to 2013" is supported if we compare the calculated mean for alcohol in 2013 (324p) with the mean from 2009 (307p). Since the difference is indeed 17p, this indicates an increase without breaching the null hypothesis, representing a significant change in spending behavior.
Step 4
State with a reason the test result supports the following statements: 14 (b)(ii)
Answer
The statement regarding the change in the mean UK consumption of alcohol per adult per week from 2009 to 2013 cannot be conclusively accepted without the statistical context of the test results. Although there was a change, without the actual evidence that definitively supports or refutes this based on calculated values, it remains indeterminate.