Answer:
<em>(a) A 99% confidence interval for the actual mean noise level in hospitals is </em><em>(44.02 db, 49.98 db)</em><em>.
</em>
<em>(b) We can be 90% confident that the actual mean noise level in hospitals is </em><em>47 db</em><em> with a margin of error of </em><em>1.89 db</em><em>.
</em>
<em>(c) Unless our sample (of 81 hospitals) is among the most unusual 2% of samples, the actual mean noise level in hospitals is between </em><em>44.41 db and 49.59 db</em><em>.
</em>
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Step-by-step explanation:
<em>The problem is incomplete. The questions are:</em>
<em />
<em>(a) A 99% confidence interval for the actual mean noise level in hospitals is </em><em>(44.02 db, 49.98 db)</em><em>.
</em>
For a 99% CI, the value of z is z=2.58
Then, the confidence interval for the mean is:
![M-z\sigma/\sqrt{n}\leq\mu\leq M-z\sigma/\sqrt{n}\\\\47-2.58*10/\sqrt{75} \leq\mu\leq47+2.58*10/\sqrt{75}\\\\47-2.98\leq\mu\leq47+2.98\\\\44.02\leq\mu\leq 49.98](https://tex.z-dn.net/?f=M-z%5Csigma%2F%5Csqrt%7Bn%7D%5Cleq%5Cmu%5Cleq%20M-z%5Csigma%2F%5Csqrt%7Bn%7D%5C%5C%5C%5C47-2.58%2A10%2F%5Csqrt%7B75%7D%20%20%5Cleq%5Cmu%5Cleq47%2B2.58%2A10%2F%5Csqrt%7B75%7D%5C%5C%5C%5C47-2.98%5Cleq%5Cmu%5Cleq47%2B2.98%5C%5C%5C%5C44.02%5Cleq%5Cmu%5Cleq%2049.98)
<em>(b) We can be 90% confident that the actual mean noise level in hospitals is </em><em>47 db</em><em> with a margin of error of </em><em>1.89 db</em><em>.
</em>
For a 90% CI, the value of z is z=1.64.
Then, we can calculate the margin of error as:
![E=z*\sigma/\sqrt{n}=1.64*10/\sqrt{75}=1.89](https://tex.z-dn.net/?f=E%3Dz%2A%5Csigma%2F%5Csqrt%7Bn%7D%3D1.64%2A10%2F%5Csqrt%7B75%7D%3D1.89)
<em>(c) Unless our sample (of 81 hospitals) is among the most unusual 2% of samples, the actual mean noise level in hospitals is between </em><em>44.41 db and 49.59 db</em><em>.
</em>
The 2% tails data corresponds, in the standard normal distirbution, to the values of z whose absolute value is higher than 2.33.
The values of db for these critical values are:
![X_1=M+z_1*\sigma/\sqrt{n}=47+(-2.33)*10/\sqrt{81}=47-2.59=44.41\\\\\\ X_2=M+z_2*\sigma/\sqrt{n}=47+(2.33)*10/\sqrt{81}=47+2.59=49.59](https://tex.z-dn.net/?f=X_1%3DM%2Bz_1%2A%5Csigma%2F%5Csqrt%7Bn%7D%3D47%2B%28-2.33%29%2A10%2F%5Csqrt%7B81%7D%3D47-2.59%3D44.41%5C%5C%5C%5C%5C%5C%20X_2%3DM%2Bz_2%2A%5Csigma%2F%5Csqrt%7Bn%7D%3D47%2B%282.33%29%2A10%2F%5Csqrt%7B81%7D%3D47%2B2.59%3D49.59)