The continuous random variable $X$ is normally distributed with $\sim \text{N}(\mu, 4^2)$ . A random sample of 12 observations of $X$ is taken and $xˉ\bar{x}$ denotes the sample mean.

Show that a 95% confidence interval for $μ\mu$ , in terms of $xˉ\bar{x}$ , is given by $(xˉ−2.26,xˉ+2.26)(\bar{x} - 2.26, \bar{x} + 2.26)$ , correct to two decimal places.

The continuous random variable $Y$ is normally distributed with $\sim \text{N}(\mu, 2^2)$ . A random sample of 15 observations of $Y$ is taken and $yˉ\bar{y}$ denotes the sample mean.

Find a 90% confidence interval for $μ\mu$ , in terms of $yˉ\bar{y}$ .

Given that $X$ and $Y$ are independent, (i) find the distribution of $Xˉ−Yˉ\bar{X} - \bar{Y}$ ; (ii) calculate the probability that the two confidence intervals from part (a) and part (b) do not overlap.

The continuous random variable $X$ is normally distributed with $\sim \text{N}(\mu, 4^2)$ . A random sample of 12 observations of $X$ is taken and $xˉ\bar{x}$ denotes the sample mean.

Show that a 95% confidence interval for $μ\mu$ , in terms of $xˉ\bar{x}$ , is given by $(xˉ−2.26,xˉ+2.26)(\bar{x} - 2.26, \bar{x} + 2.26)$ , correct to two decimal places.

The continuous random variable $Y$ is normally distributed with $\sim \text{N}(\mu, 2^2)$ . A random sample of 15 observations of $Y$ is taken and $yˉ\bar{y}$ denotes the sample mean.

Find a 90% confidence interval for $μ\mu$ , in terms of $yˉ\bar{y}$ .

3.1 The Normal Distribution

Question 5

Markscheme

3.1 The Normal Distribution

Question 5

Markscheme

3.1 The Normal Distribution

Question 5

3.1 The Normal Distribution Questions

3.1 The Normal Distribution

Question 5

3.1 The Normal Distribution Questions