It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A&E) patients. After some new initiatives were introduced, a random sample of 12 patients from the hospital's A&E Department had the following waiting times, in hours. 4.25 3.90 4.15 3.95 4.20 4.50 5.00 3.85 4.25 4.05 3.80 3.95 Carry out a hypothesis test at the 10% significance level to investigate whether the mean waiting time at this hospital’s A&E department has changed. You may assume that the waiting times are normally distributed with standard deviation 0.8 hours.

A-Level AQA Maths Mechanics Quantities, Units & Modelling: It is known that a hospital has

It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A&E) patients - AQA - A-Level Maths Mechanics - Question 14 - 2020 - Paper 3

Question 14

It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A&E) patients. After some new initiatives were introduced, a random s... show full transcript

Worked Solution & Example Answer:It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A&E) patients - AQA - A-Level Maths Mechanics - Question 14 - 2020 - Paper 3

Step 1

State both hypotheses correctly for two-tailed test

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Answer

The null hypothesis (H0) states that the mean waiting time is equal to 4 hours, i.e., $ar{ ext{H}}_0: u = 4$ . The alternative hypothesis (H1) states that the mean waiting time has changed, i.e., $H_1: u eq 4$ .

Step 2

Calculate means of the sample

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Answer

The sample waiting times are: [4.25, 3.90, 4.15, 3.95, 4.20, 4.50, 5.00, 3.85, 4.25, 4.05, 3.80, 3.95]. The mean ( $ar{x}$ ) can be calculated as follows:

$ar{x} = rac{4.25 + 3.90 + 4.15 + 3.95 + 4.20 + 4.50 + 5.00 + 3.85 + 4.25 + 4.05 + 3.80 + 3.95}{12} = 4.125$

Step 3

Formulate the test statistic

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Answer

Using the formula for the test statistic:

u_0}{ rac{ ext{SD}}{ ext{sqrt}(n)}}$$ where $ar{x} = 4.125$, $ u_0 = 4$, $ ext{SD} = 0.8$, and $n = 12$. Substituting the values: $$t = rac{4.125 - 4}{ rac{0.8}{ ext{sqrt}(12)}} \\ = rac{0.125}{ rac{0.8}{3.464}} \\ = rac{0.125}{0.230} \\ ext{approximately} = 0.543$$

Step 4

Obtain the critical value

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Answer

For a two-tailed test at the 10% significance level, the critical values are found using the t-distribution table. With $df = n-1 = 11$ , the critical values are approximately $ext{±}1.65$ .

Step 5

Compare the test statistic to critical value

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Answer

The calculated test statistic is $0.543$ which falls within the range of the critical values $[-1.65, 1.65]$ . Therefore, we fail to reject the null hypothesis.

Step 6

Conclude correctly in context

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Answer

There is insufficient evidence to suggest that the mean waiting time at this hospital's A&E department has changed.

It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A&E) patients - AQA - A-Level Maths Mechanics - Question 14 - 2020 - Paper 3

Worked Solution & Example Answer:It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A&E) patients - AQA - A-Level Maths Mechanics - Question 14 - 2020 - Paper 3

State both hypotheses correctly for two-tailed test

Calculate means of the sample

Formulate the test statistic

Obtain the critical value

Compare the test statistic to critical value

Conclude correctly in context

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