The number of working hours per week of employees in a company is modelled by a normal distribution with mean of 34 hours and a standard deviation of 4.5 hours - AQA - A-Level Maths Pure - Question 17 - 2022 - Paper 3

We will use the sample mean ( \bar{x} ), the population mean ( \mu ), the standard deviation ( \sigma ), and the sample size ( n ).

Using the formula for the test statistic:

$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

Substituting the values: $z = \frac{36.2 - 34}{4.5 / \sqrt{30}} =\frac{2.2}{0.8202} \approx 2.68$

For a one-tailed test at the 2.5% significance level, we find the critical z-value using a z-table.

The critical value for \alpha = 0.025 is approximately 1.96.

Compare the calculated z-value with the critical value:

Since 2.68 > 1.96, we reject the null hypothesis.

There is sufficient evidence to suggest that the mean working hours have increased.