Step 1

(a)

The known values are,

\(\displaystyle\sigma={230}\),

\(\displaystyle{n}={50}\)

Critical value:

By using the z-tables, the critical value at \(5\%\) level of significance for a two tailed z-distribution is,

\(\displaystyle{Z}_{{{0.05}\text{/}{2}}}=\pm{1.96}\)

The value of margin of error is

\(\displaystyle{m}={z}_{{\alpha\text{/}{2}}}{\left(\frac{\sigma}{\sqrt{{n}}}\right)}\)

\(\displaystyle={1.96}{\left(\frac{230}{\sqrt{{50}}}\right)}\)

\(\displaystyle={1.96}\times{32.52691}\)

\(\displaystyle={63.75275}\)

\(\displaystyle\approx{63.75}\)

Thus, the required margin of error is \(\displaystyle{m}={63.75}\)

Step 2

(b)

The sample mean is,

\(\displaystyle\overline{{x}}=\frac{{{\sum_{{{i}-{1}}}^{{n}}}{x}_{{i}}}}{{n}}\)

\(\displaystyle=\frac{{{1902}+{2042}+{1936}+{1817}+\ldots+{2232}+{2294}}}{{50}}\)

\(\displaystyle=\frac{91842}{{50}}\)

\(\displaystyle={1837}\)

The \(95\%\) confidence interval for the mean cost in dollars of the first year of owning and caring for an Irish Red and White Setter is,

\(\displaystyle{C}{I}=\overline{{x}}\pm{m}\)

\(\displaystyle={1837}\pm{63.75}\)

\(\displaystyle={\left({1837}-{63.75},{1837}+{63.75}\right)}\)

\(\displaystyle={\left({1773.25},{1900.75}\right)}\)

Thus, the required confidence interval is, ($1773.25 to $1900.75)

(a)

The known values are,

\(\displaystyle\sigma={230}\),

\(\displaystyle{n}={50}\)

Critical value:

By using the z-tables, the critical value at \(5\%\) level of significance for a two tailed z-distribution is,

\(\displaystyle{Z}_{{{0.05}\text{/}{2}}}=\pm{1.96}\)

The value of margin of error is

\(\displaystyle{m}={z}_{{\alpha\text{/}{2}}}{\left(\frac{\sigma}{\sqrt{{n}}}\right)}\)

\(\displaystyle={1.96}{\left(\frac{230}{\sqrt{{50}}}\right)}\)

\(\displaystyle={1.96}\times{32.52691}\)

\(\displaystyle={63.75275}\)

\(\displaystyle\approx{63.75}\)

Thus, the required margin of error is \(\displaystyle{m}={63.75}\)

Step 2

(b)

The sample mean is,

\(\displaystyle\overline{{x}}=\frac{{{\sum_{{{i}-{1}}}^{{n}}}{x}_{{i}}}}{{n}}\)

\(\displaystyle=\frac{{{1902}+{2042}+{1936}+{1817}+\ldots+{2232}+{2294}}}{{50}}\)

\(\displaystyle=\frac{91842}{{50}}\)

\(\displaystyle={1837}\)

The \(95\%\) confidence interval for the mean cost in dollars of the first year of owning and caring for an Irish Red and White Setter is,

\(\displaystyle{C}{I}=\overline{{x}}\pm{m}\)

\(\displaystyle={1837}\pm{63.75}\)

\(\displaystyle={\left({1837}-{63.75},{1837}+{63.75}\right)}\)

\(\displaystyle={\left({1773.25},{1900.75}\right)}\)

Thus, the required confidence interval is, ($1773.25 to $1900.75)