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A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes - Edexcel - A-Level Maths Statistics - Question 5 - 2020 - Paper 1
Question 5
A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 ... show full transcript
Worked Solution & Example Answer:A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes - Edexcel - A-Level Maths Statistics - Question 5 - 2020 - Paper 1
Step 1
Find the probability that the time spent with a randomly selected patient is more than 15 minutes.
Answer
Using the normal distribution model, we have the mean, ( \mu = 10 ) minutes, and standard deviation, ( \sigma = 4 ) minutes. We want to find ( P(T > 15) ).
First, we standardize this value:
[ Z = \frac{X - \mu}{\sigma} = \frac{15 - 10}{4} = 1.25 ]
Now, we find ( P(Z > 1.25) ) using Z-tables or a calculator:
[ P(Z > 1.25) \approx 0.1064 ] (approximately 0.106).
Step 2
Stating your hypotheses clearly and using a 5% significance level, test whether or not there is evidence to support the patients’ complaint.
Answer
Let:\n( H_0: \mu = 10 ) (the mean time is 10 minutes)\n( H_1: \mu > 10 ) (the mean time is greater than 10 minutes)\n\nThe sample mean ( \bar{x} = 11.5 ) with sample size ( n = 20 ). We calculate the test statistic:
[ Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} = \frac{11.5 - 10}{\frac{4}{\sqrt{20}}} \approx 2.35 ]
Using a Z-table, the critical value for a one-tailed test at 5% significance level is approximately 1.645. Since 2.35 exceeds 1.645, we reject ( H_0 ). This indicates evidence to support the patients’ complaint.
Step 3
Step 4
Step 5
Hence explain why this normal distribution may not be a good model for T.
Answer
The normal distribution assumes that values can take any real number, including negatives. However, since time cannot be negative, this model leads to non-negligible probabilities for negative values (e.g., ( P(T < 0) )). Therefore, it is inappropriate as it suggests an unrealistic scenario.
Step 6
Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place.
Answer
The suggested new model only includes values of ( T > 2 ). For a normal distribution, the median is equal to the mean. So, we need to find the median of a truncated normal distribution above 2.
With the mode being at 5, the median would adjust slightly. Assuming we find it in the truncated range, we have:
Median = ( 5.9 ).