Question

A university wants to compare out-of-state applicants' mean SAT math scores (μ1) to in-state applicants' mean SAT math scores (μ2). The university looks at 35 in-state applicants and 35 out-of-state applicants. The mean SAT math score for in-state applicants was 540, with a standard deviation of 20. The mean SAT math score for out-of-state applicants was 555, with a standard deviation of 25. It is reasonable to assume the corresponding population standard deviations are equal. To calculate the confidence interval for the difference μ1 − μ2, what is the number of degrees of freedom of the appropriate probability distribution?

64

64.87

68

69

Answer 1

`df=n_1 +n_2 -1=35+35-2=68`

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X_1 =540` represent the sample mean 1 (in-state applicants)

`\bar X_2 =555` represent the sample mean 2 (out-of state applicants)

n1=35 represent the sample 1 size (in-state applicants)

n2=35 represent the sample 2 size (out-of state applicants)

`s_1 =20` sample standard deviation for sample 1 (in-state applicants)

`s_2 =25` sample standard deviation for sample 2 (out-of state applicants)

`\mu_1 -\mu_2` parameter of interest.

We are assuming that the population standard deviations are equal.

The confidence interval for the difference of means is given by the following formula:

`(\bar X_1 -\bar X_2) \pm t_(\alpha/2)s_p \sqrt{(1)/(n_1)+(1)/(n_2)}` (1)

Where

`s^2=((n_1 -1)s_1^2 +(n_2-1)s_2^2)/(n_1 +n_2 -2)`

`s^2=((35 -1)20^2 +(35-1)25^2)/(35 +35 -2)=512.5`

`s=√(512.5)=22.638`

The point of estimate for
`\mu_1 -\mu_2` is just given by:

`\bar X_1 -\bar X_2 =540-555=-15`

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n_1 +n_2 -1=35+35-2=68`