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 σ = $255.

1770 1902
2052 2042
1379 1936
1939 1817
1854 1504
1913 1572
2163 1532
1737 1907
1888 1882
1737 2153
2230 1945
2131 1335
1813 2006
2118 1516
1978 1839
2166 1739
1482 1456
1700 1798
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)?
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

(A) The margin of error = 70.68.

(B) 95% confidence interval for the mean cost of the first year of owning and caring for an Irish Red and White Setter is [$1766.16, $1907.52].

Explanation:

We are given that 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). A sample of 50 will be used.

Based on past studies, the population standard deviation is assumed known with σ = $255.

Let
`\mu` = population mean cost = $1,843

`\sigma` = population standard devaition = $255

n = sample size = 50

(A) The Margin of error formula for a 95% confidence interval of the mean cost of the first year of owning and caring for this breed is given by;

The margin of error =
`Z_(_(\alpha)/(2)_) * (\sigma)/(√(n) )`

Here,
`\alpha` = level of significance = 1 - 0.95 = 5%

And the critical value of z at 2.5% (
`(0.05)/(2)`) level of significance is given as 1.96.

So, margin of error =
`1.96 * (255)/(√(50) )` = 70.68.

(B) We are given with the data collected from fifty owners of Irish Setters on the cost of the first year of owning and caring for their dogs below;

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.

Firstly, we have to calculate the mean of this above data, i.e;

Sample Mean,
`\bar X` =
`(\sum X_i)/(n)` =
`(91842)/(50)` = 1836.84

95% confidence interval for the mean cost (
`\mu`) of the first year of owning and caring for an Irish Red and White Setter is given by;

=
`\bar X \pm \text{Margin of error}`

= [1836.84 - 70.68 , 1836.84 + 70.68]

= [1766.16 , 1907.52]

Hence, 95% confidence interval for the mean cost of the first year of owning and caring for an Irish Red and White Setter is [$1766.16 , $1907.52].