Suppose you take independent random samples from populations with means mu1 and mu2 and standard deviations sigma1 and sigma2. Furthermore, assume eit

Question

Suppose you take independent random samples from populations with means $μ 1 and μ 2$ and standard deviations $σ 1 and σ 2$ . Furthermore, assume either that (i) both populations have normal distributions, or (ii) the sample sizes ( $n_{1} and n_{2}$ ) are large. If $X_{1} and X_{2}$ are the random sample means, then how does the quantity
$\frac{(\overset{―}{x_{1}} - \overset{―}{x_{2}}) - (μ_{1} - μ_{2})}{\sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}}$
Give the name of the distribution and any parameters needed to describe it.

Answer 1

Step 1
Given:
Two independent random samples from populations with
means-

μ_{1} and μ_{2}

standard deviations -

σ_{1} and σ_{2}

Assumptions:
i.Both populations have normal distributions
ii.The sample sizes(

n_{1} and n_{2}

) are large
Step 2 answer

X_{1} and X_{2}

are random samples then the quantity

t (s a y) = \frac{(\overset{―}{x_{1}} - \overset{―}{x_{2}}) - (μ_{1} - μ_{2})}{\sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}}

this quantity

t (s a y) \sim N (0, 1) \Rightarrow

quantity t has standard normal distribution with mean zero and variance 1 the parameters are

μ_{1} - μ_{2}

is the difference between the population means and

σ_{1} and σ_{2}

are the population standard deviations.

Suppose you take independent random samples from populations with means mu1 and mu2 and standard deviations sigma1 and sigma2. Furthermore, assume eit

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