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Pavan V Parekh · Answer 1 · 2021-02-26T00:31:29+0000

Answer:

$P(\bar X < 22)$

And we can use the z score given by:

$z= (\bar X -\mu)/((\sigma)/(√(n)))$

And replacing we got:

z = (22-30)/((8)/(√(2)))= -2

So we can rewrite the probability like this:

$P(\bar X < 22)= P(Z<-2)$

And using the normal standard distribution or excel we got:

$P(\bar X < 22)= P(Z<-2)=0.0228$

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:

$X \sim N(30,8)$

Where
$\mu=30$ and
$\sigma=8$

We select a sample size of n =4 and since the distribution for X is normal, then the distribution for the sample mean
$\bar X$ is given by:

$\bar X \sim N(\mu, (\sigma)/(√(n)))$

And we want this probability: