Question

A recent survey compared the costs of adoption through public and private agencies. For a sample of 16 adoptions through a public agency, the mean cost was $21,045, with a standard deviation of $835. For a sample of 18 adoptions through a private agency, the mean cost was $22,840, with a standard deviation of $1,545. Can we conclude the mean cost is larger for adopting children through a private agency? Use the .05 significance level.

Find the degrees of freedom for unequal variance test. (Round down your answer to the nearest whole number)

Answer 1

The degrees of freedom for the unequal variance test is 29.

The degrees of freedom for the unequal variance t-test can be calculated using the following formula:

`[ df = \frac{{\left( \frac{{s_x^2}}{{n_x}} + \frac{{s_y^2}}{{n_y}} \right)^2}}{{\frac{{s_x^4}}{{n_x^2(n_x-1)}} + \frac{{s_y^4}}{{n_y^2(n_y-1)}}}} ]`

Where:

`( s_x )` and
`( s_y )` are the standard deviations of the two samples

`( n_x )` and
`( n_y )` are the sizes of the two samples

Using the given data:

For the public agency sample:
`( s_x = 835 )` and
`( n_x = 16 )`

For the private agency sample:
`( s_y = 1545 )` and
`( n_y = 18 )`

Substituting these values into the formula, we get:

`[ df = \frac{{\left( \frac{{835^2}}{{16}} + \frac{{1545^2}}{{18}} \right)^2}}{{\frac{{835^4}}{{16^2(16-1)}} + \frac{{1545^4}}{{18^2(18-1)}}}} ]`

`[ df \approx 29.78 ]`

Rounding down to the nearest whole number, the degrees of freedom for the unequal variance t-test is 29.

Therefore, the degrees of freedom for the unequal variance test is 29.