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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 lifetim... 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 the relation for median M:
Answer
Given that the logarithm of the lifetime ( (\log_{10} X)) follows a normal distribution, the median of (X) will be (M = 10^{\mu}), where (\mu) is the mean of the distribution. Here, (\mu = 1.5), hence:
Thus, the median lifetime is approximately 31.62 days.
Step 2
Find also P(X > 50):
Answer
To find (P(X > 50)), we need to convert this into the corresponding logarithmic expression:
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Calculate (\log_{10}(50)):
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We can then standardize this value to use the properties of the normal distribution: (Z = \frac{\log_{10}(50) - \mu}{\sigma} = \frac{1.699 - 1.5}{0.2} = 0.995)
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Now, we use the cumulative distribution function (CDF) for the standard normal distribution:
Thus, (P(X > 50) \approx 0.16).