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The waiting time, \(T\) hours, to see a particular doctor at a clinic has a mean of 0.5 hours and a standard deviation of 0.3 hours.
A sample of 35 waiting times is chosen at random.
Use the standard normal distribution table (provided) to find the probability that the average waiting time of the sample of 35 patients is between 0.43 hours and 0.50 hours. Give your answer as a percentage correct to 1 decimal place. (3 marks)
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\(\Pr(0.43<\overline{T}<0.50) \approx 41.6\%\)
\(\text{Sample size is > 30} \ \ \Rightarrow \ \ \text{Central Limit Theorem applies}\)
\(\text{The sample mean,}\ \overline{T}, \text{for random samples of size 35 is}\)
\(\text{approximately normally distributed, where:}\)
\(\mu=0.5\ \ \text{and}\ \ \sigma=\dfrac{0.3}{\sqrt{n}}=\dfrac{0.3}{\sqrt{35}} \approx 0.0507\)
\(\text{By the central limit theorem:}\)
\(Z=\dfrac{\overline{T}-0.5}{\frac{0.3}{\sqrt{35}}} \sim N(0,1)\)
| \(\Pr(0.43<\overline{T}<0.50)\) | \(=\Pr\left(\dfrac{0.43-0.50}{0.0507}<Z<\dfrac{0.50-0.50}{0.0507}\right)\) |
| \(=\Pr(-1.38<Z<0)\) | |
| \(=\Pr(0<Z<1.38)\) | |
| \(=0.9162-0.5000\) | |
| \(=0.4162 = 41.6\%\ \text{(1 d.p.)}\) |