The random variable $X$ has the distribution $X \sim \mathrm{N}(\mu, \sigma^2)$. It is given tha

Question

The random variable $X$ has the distribution $X \sim \mathrm{N}(\mu, \sigma^2)$. It is given that $X_1$ and $X_2$ are $2$ independent observations of $X$ with $\mathrm{P}(4 < X < 6) = 0.5$ and $\mathrm{P}(X_1 + X_2 > 10) = 0.5$. Show that $\mu = 5$ and state, with a reason, the value of $\mathrm{P}(2X > 10)$.[3] Find the value of $\sigma$, giving your answer to $5$ significant figures.[3] A random sample of $n$ observations of $X$ is obtained. Given that there is a probability of more than $95\%$ such that the sample mean is within $0.1$ units of $\mu$, find the least value of $n$.[4]

Accepted Answer

(i) $\mu=5$, $\mathrm P(2X>10)=0.5$ (ii) $\sigma=1.4826$ (iii) least $n=845$

Sampling - Determining Sample Size from Probability Conditions

Sampling - Determining Sample Size from Probability Conditions

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