Internal auditors are often used to review an organization's financial statements such as balance sheets, income statements, and cash flow statements prior to public filings. Auditors seek to verify that the financial statements accurately represent the financial position of the organization and that the statements follow accepted accounting principles. Many errors that are discovered by auditors are minor errors that are easily corrected. However, some errors are serious and require substantial time to rectify. Suppose that the financial statements of $567$ public companies are audited. The file internalaudit$4$ contains the number of errors discovered during the internal audit of each of these $567$ public companies that were classified as "serious." Use the data in the file internalaudit$4$ to answer the following.

$($a$)$

Construct an empirical discrete probability distribution for the number of serious errors discovered during the internal audits of these $567$ public companies. $($Round your answers to five decimal places.$)$

Number of Serious Errors Probability

f$($x$)$

$0$

$1$

$2$

$3$

$4$

$5$

$6$

$($b$)$

What is the probability that a company has no serious errors in its financial statements? $($Round your answer to five decimal places.$)$

$($c$)$

What is the probability that a company has four or more serious errors in its financial statements? $($Round your answer to five decimal places.$)$

$($d$)$

What is the expected number of serious errors in a company's financial statements? $($Round your answer to four decimal places.$)$

$($e$)$

What is the variance of the number of serious errors in a company's financial statements? $($Round your answer to four decimal places.$)$

$($f$)$

What is the standard deviation of the number of serious errors in a company's financial statements? $($Round your answer to four decimal places.$)$