Answer :
Answer:
[tex]df=n_1 +n_2 -1=35+35-2=68[/tex]
Step-by-step 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".
[tex]\bar X_1 =540[/tex] represent the sample mean 1 (in-state applicants)
[tex]\bar X_2 =555[/tex] 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)
[tex]s_1 =20[/tex] sample standard deviation for sample 1 (in-state applicants)
[tex]s_2 =25[/tex] sample standard deviation for sample 2 (out-of state applicants)
[tex]\mu_1 -\mu_2[/tex] 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:
[tex](\bar X_1 -\bar X_2) \pm t_{\alpha/2}s_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}[/tex] (1)
Where
[tex]s^2=\frac{(n_1 -1)s_1^2 +(n_2-1)s_2^2}{n_1 +n_2 -2}[/tex]
[tex]s^2=\frac{(35 -1)20^2 +(35-1)25^2}{35 +35 -2}=512.5[/tex]
[tex]s=\sqrt{512.5}=22.638[/tex]
The point of estimate for [tex]\mu_1 -\mu_2[/tex] is just given by:
[tex]\bar X_1 -\bar X_2 =540-555=-15[/tex]
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n_1 +n_2 -1=35+35-2=68[/tex]