Question

The average annual cost of the first year of owning and caring for a large dog is $1,843 (US News and World Report, September 9, 2013). The Irish Red and White Setter Association of America has requested a study to estimate the annual first-year cost for owners of this breed. A sample of 50 will be used. Based on past studies, the population standard deviation is assumed known with o $255. a. What is the margin of error for a 95% confidence interval of the mean cost of the first year of owning and caring for this breed (to 2 decimals)? DATA file b. The DATAfile Setters contains data collected from fifty owners of Irish Setters on the cost of the first year of owning and caring for their dogs. Use these data sets to compute the sample mean (to 2 decimals). Using this sample, what is the 95% confidence interval for the mean cost of the first year of owning and caring for an Irish Red and White Setter (to 2 decimals)?

Answer 1

$\small \mathbf{Solution}$

255 given : n= 50, a

$\small {\color{blue} \mathbf{a)}}\;\mathbf{for\; 95\% \;confidence\;level, \;z_{\frac{\alpha }{2}}=1.96}$

Margin of error (E)4 el

$\small Margin\;of\;error\;(E)=1.96*\left \{ \frac{255}{\sqrt{50}} \right \}$

$\small Margin\;of\;error\;(E)=\mathbf{{\color{Red} 70.68}}$

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$\small {\color{blue} \mathbf{b)}}\;Sample\;mean\;(\bar X)=\frac{\sum X}{n}$

1902

2042

1936

1817

1504

1572

1532

1907

1882

2153

1945

1335

2006

1516

1839

1739

1456

1958

1934

2094

1739

1434

1667

1679

1736

1670

1770

2052

1379

1939

1854

1913

2163

1737

1888

1737

2230

2131

1813

2118

1978

2166

1482

1700

1679

2060

1683

1850

2232

2294

$\small Total \;sum\;\left ( \sum X \right )=91842$

$\small \bar X=\frac{91842}{50}=\mathbf{{\color{Red} 1836.84 }}$

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$\small {\color{blue} \mathbf{c)}}\;Confidence\;Interval=\bar X \pm Margin\;of\;error\;(E)$

$\small =1836.84 \pm 70.68$

$\small =\left (1836.84 - 70.68,\;1836.84 + 70.68 \right )$

$\small \mathbf{=\left ({\color{Red} 1766.16},\;{\color{Red} 1907.52 }\right )}$