Math / Data & Statistics

Question1. Suppose $X_{1}$ is normally distributed with a mean of 4 and a standard deviation of 1 and suppose $X_{2}$ is normally distributed with a mean of 5 and a standard deviation of 2 . (a) Explain why $\bar{X}_{1}-\bar{X}_{2}$ is normally distributed, even though both $n_{1}$ and $n_{2}$ are small. (2 marks)

Studdy Solution

STEP 1

1. $X_1$ and $X_2$ are independent normal random variables.
2. The means and standard deviations are given: $\mu_1 = 4, \sigma_1 = 1$ for $X_1$ and $\mu_2 = 5, \sigma_2 = 2$ for $X_2$ .
3. We are considering the difference of sample means $\bar{X}_1 - \bar{X}_2$ .

STEP 2

1. Discuss the properties of normal distributions.
2. Explain the distribution of the difference of two independent normal variables.

STEP 3

Discuss the properties of normal distributions:
- A normal distribution is fully characterized by its mean and standard deviation. - If a random variable is normally distributed, any linear combination of independent normal random variables is also normally distributed.

STEP 4

Explain the distribution of the difference of two independent normal variables:
- Since $X_1$ and $X_2$ are independent and normally distributed, the sample means $\bar{X}_1$ and $\bar{X}_2$ are also normally distributed. - The difference $\bar{X}_1 - \bar{X}_2$ is a linear combination of independent normal variables. - Therefore, $\bar{X}_1 - \bar{X}_2$ is normally distributed regardless of the sample sizes $n_1$ and $n_2$ .
The explanation shows that $\bar{X}_1 - \bar{X}_2$ is normally distributed due to the properties of normal distributions and the independence of $X_1$ and $X_2$ .

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