The mean annual income for people in a certain city (in thousands of dollars) is 42, with a standard deviation of 30. A pollster draws a sample of 90 people to interview. a. What is the probability that the sample mean income is less than 38?

And we can use the z score formula given by:

And if we find the z score for 38 we got:

So then we want to find this probability:

And we can use the normal standard distribution or excel and we got:

Step-by-step explanation:

For this case we define the random variable X as the annual income for people at certain city. And we know the following properties:

They select a sample size of n = 90>30. So then we can assume that the central limit can be applied.

From the central limit theorem we know that the distribution for the sample mean is given by:

We want to calculate this probability:

And we can use the z score formula given by:

And if we find the z score for 38 we got:

So then we want to find this probability:

And we can use the normal standard distribution or excel and we got:

this shows a reflection over the x-axis and is translated one unit down and 6 units to the right (x + 6, y - 1)

hope this !

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step-by-step explanation: