A survey during 2013 investigated mean expenditure on bread and on alcohol - AQA - A-Level Maths Pure - Question 14 - 2019 - Paper 3

To carry out the hypothesis test, we start by stating the null and alternative hypotheses:

• Null Hypothesis, $H_0: \mu = 123$ (the mean expenditure per adult on bread has not changed)
• Alternative Hypothesis, $H_1: \mu \neq 123$ (the mean expenditure per adult on bread has changed)

Next, we calculate the test statistic using the formula:

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

Where:

• $\bar{x} = 127$ (sample mean)
• $\mu_0 = 123$ (hypothesized population mean)
• $s = 70$ (standard deviation)
• $n = 12144$ (sample size)

Substituting the values into the formula gives:

$t = \frac{127 - 123}{70 / \sqrt{12144}} = \frac{4}{70 / 110.25} = \frac{4 \times 110.25}{70} \approx 6.30$

Now, we find the critical values for a two-tailed test at the 5% significance level. Looking up values from the t-distribution table or using a calculator, we find that the critical values are approximately ±1.96.

Since the calculated test statistic (6.30) is greater than 1.96, we reject the null hypothesis. This indicates that there is evidence to suggest that the mean expenditure per adult per week on bread changed from 2012 to 2013.