A dentist knows from past records that 10% of customers arrive late for their appointment - Edexcel - A-Level Maths Statistics - Question 4 - 2022 - Paper 1

The alternative hypothesis ( H_1) reflects the manager's belief that there has been a change in the arrival pattern, indicating that the proportion is not equal to 10%. Therefore, we state this as:

Given a significance level of 5%, we need to find the critical regions for a two-tailed test. Since we are dealing with a sample size of 50, we can use the standard normal distribution for our critical values.

To find the critical values, we can refer to a z-table or use software to determine the Z-scores that correspond to the upper and lower 2.5% of the distribution (for a total of 5%). The critical region corresponds to:

• Lower critical value: $Z < -1.96$
• Upper critical value: $Z > 1.96$

In this case, we can calculate the actual level of significance using the observed data. The proportion of customers arriving late in this sample is:

ext{Sample proportion} = rac{15}{50} = 0.3

Since $0.3$ is outside the critical region determined in part (b) (which was $Z < -1.96$ or $Z > 1.96$), we need to calculate the actual test statistic:

This can also help in determining the p-value associated with it, confirming that the level of significance is approximately $0.0297$.

Given that the observed value of 15 late appointments falls within the critical region defined earlier, we have sufficient evidence to reject the null hypothesis. This suggests that the actual proportion of customers arriving late is significantly different from 10%. Therefore, we can conclude that the manager's belief in a change in the proportion of late arrivals is supported by the data.