A sample of 400 high school students showed that they spend an average of 70 min
ID: 3159265 • Letter: A
Question
A sample of 400 high school students showed that they spend an average of 70 minutes a day watching television with a standard deviation of 14 minutes. Another sample of 500 college students showed that they spend an average of 55 minutes a day watching television with a standard deviation of 12 minutes. Construct a 99% confidence interval for the difference between the mean times spent watching television by all high school students and all college students. Test at the 2.5% significance level if the mean time spent watching television per day by high school students is higher than the mean time spent watching television by college students.Explanation / Answer
a)
Calculating the means of each group,
X1 = 70
X2 = 55
Calculating the standard deviations of each group,
s1 = 14
s2 = 12
Thus, the standard error of their difference is, by using sD = sqrt(s1^2/n1 + s2^2/n2):
n1 = sample size of group 1 = 400
n2 = sample size of group 2 = 500
Also, sD = 0.882043083
For the 0.99 confidence level, then
alpha/2 = (1 - confidence level)/2 = 0.005
z(alpha/2) = 2.575829304
Thus,
lower bound = [X1 - X2] - z(alpha/2) * sD = 12.72800758
upper bound = [X1 - X2] + z(alpha/2) * sD = 17.27199242
Thus, the confidence interval is
( 12.72800758 , 17.27199242 ) [ANSWER]
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b)
Formulating the null and alternative hypotheses,
Ho: u1 - u2 <= 0
Ha: u1 - u2 > 0
At level of significance = 0.025
As we can see, this is a right tailed test.
Calculating the means of each group,
X1 = 70
X2 = 55
Calculating the standard deviations of each group,
s1 = 14
s2 = 12
Thus, the standard error of their difference is, by using sD = sqrt(s1^2/n1 + s2^2/n2):
n1 = sample size of group 1 = 400
n2 = sample size of group 2 = 500
Also, sD = 0.882043083
Thus, the z statistic will be
z = [X1 - X2 - uD]/sD = 17.00597203
where uD = hypothesized difference = 0
Now, the critical value for z is
zcrit = 1.96
Also, using p values, as this is right tailed,
p = 3.70824E-65
As z > 1.96, and P < 0.025, WE REJECT THE NULL HYPOTHESIS.
There is significant evidence at 0.025 level that the mean time spent watching television per day by high school students is higher than that of college students. [CONCLUSION]