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The lifetime, X days, of a particular insect is such that log_{10} X has a normal distribution with mean 1.5 and standard deviation 0.2 - CIE - A-Level Further Maths - Question 6 - 2010 - Paper 1
Question 6
The lifetime, X days, of a particular insect is such that log_{10} X has a normal distribution with mean 1.5 and standard deviation 0.2. Find the median lifetime. F... show full transcript
Worked Solution & Example Answer:The lifetime, X days, of a particular insect is such that log_{10} X has a normal distribution with mean 1.5 and standard deviation 0.2 - CIE - A-Level Further Maths - Question 6 - 2010 - Paper 1
Step 1
Find relation for median M
Answer
To find the median lifetime, we first acknowledge the properties of the log-normal distribution. Since we are given that ( \log_{10} X ) has a normal distribution, the median of X, denoted as M, can be related to its logarithm by:
[ M = 10^{\mu} ]
where ( \mu = 1.5 ) is the mean of ( \log_{10} X ).
Thus, substituting in:
[ M = 10^{1.5} \approx 31.62 ]
Step 2
Find also P(X > 50)
Answer
To find ( P(X > 50) ), we relate it to the normal distribution of ( \log_{10} X ):
First, we convert the value 50: [ \log_{10}(50) \approx 1.699 ]
Hence, we need to find: [ P(X > 50) = P(\log_{10} X > \log_{10}(50)) ]
Standardizing this: [ P(\log_{10} X > 1.699) = 1 - \Phi\left( \frac{\log_{10}(50) - 1.5}{0.2} \right) ] [ = 1 - \Phi\left( \frac{1.699 - 1.5}{0.2} \right) ] [ = 1 - \Phi(0.995) \approx 0.160 ]