a)

The formula for estimation is:

\(\displaystyle\mu={M}\pm{Z}{\left({s}_{{M}}\right)}where:\)

\(\displaystyle{M}=\) sample mean

\(\displaystyle{Z}={Z}\) statistic determined by confidence level

\(\displaystyle{s}_{{M}}=\text{standard error}=\sqrt{{{\left({s}^{2}\text{/}{n}\right)}}}\)

Step 2

Calculation

\(\displaystyle{M}={205}\)

\(\displaystyle{Z}={2.58}\)

\(\displaystyle{s}_{{M}}=\sqrt{{{\left({7}^{2}\text{/}{20}\right)}}}={1.57}\)

\(\displaystyle\mu={M}\pm{Z}{\left({s}_{{M}}\right)}\)

\(\displaystyle\mu={205}\pm{2.58}\cdot{1.57}\)

\(\displaystyle\mu={205}\pm{4.03}\)

Result

\(\displaystyle{M}={205},{99}\%{C}{I}{\left[{200.97},{209.03}\right]}\)

Part b

Z Score Calculations

\(\displaystyle{Z}={\left({M}-\mu\right)}\text{/}\sqrt{{{\left(\sigma^{2}\text{/}{n}\right)}}}\)

\(\displaystyle{Z}={\left({295}-{300}\right)}\text{/}\sqrt{{{\left({8}\text{/}{20}\right)}}}\)

\(\displaystyle{Z}=-{5}\text{/}{0.63246}\)

\(\displaystyle{Z}=-{7.90569}\)

The value of z is -7.90569. The value of p is \(\displaystyle<{.00001}\). The result is significant at p \(\displaystyle<{.05}\)