Taxes: The Internal Revenue Service reports that the mean federal income tax paid in the year 2010 was $\$ 8040$. Assume that the standard deviation is $\$ 4800$. The IRS plans to draw a sample of 1000 tax returns to study the effect of a new tax law.
Part 1 of 5
(a) What is the probability that the sample mean tax is less than $\$ 8100$ ? Round the answer to at least four decimal places.
The probability that the sample mean tax is less than $\$ 8100$ is 0.6537 .
Part: $1 / 5$
Part 2 of 5
(b) What is the probability that the sample mean tax is between $\$ 7400$ and $\$ 8000$ ? Round the answer to at least four decimal places.
The probability that the sample mean tax is between $\$ 7400$ and $\$ 8000$ is
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Step 1 ：The problem is asking for the probability that the sample mean tax is less than $8100. This is a problem of normal distribution.

Step 2 ：The mean of the sample mean distribution is the same as the population mean, which is $8040.

Step 3 ：The standard deviation of the sample mean distribution is the population standard deviation divided by the square root of the sample size, which is $4800 / \sqrt{1000}$.

Step 4 ：We can then calculate the z-score for $8100, which is ($8100 - $8040) / ($4800 / \sqrt{1000}).

Step 5 ：The probability that the sample mean tax is less than $8100 is the cumulative distribution function (CDF) of the normal distribution at this z-score.

Step 6 ：Final Answer: The probability that the sample mean tax is less than $8100 is \(\boxed{0.6537}\).