Some accounting firms give the client an option to pay an additional fee when doing their tax return, that guarantees company support should the client be audited. A large accounting firm would like to update their fee for this type of service. In previous years, the average cost to the firm for being involved in a client audit was $$680.$ To determine if this cost has changed, the firm randomly samples $32$ clients who got audited. The average audit cost for this sample of clients was $$713$ with a standard error of $$15\mathrm{.}20.$

$($a$)$ Compute a $90\%$ confidence interval for the true average involvement cost for all client audits for this company: $($Use $4$ decimals.$)$

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$($b$)$ Based on the above interval, should the firm conclude the average audit involvement cost is different than $$680?$

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$,$ there is

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evidence the true average cost is

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$$680,$ with $90\%$ confidence