3 Two hundred students are enrolled in an Economics class Af

3. Two hundred students are enrolled in an Economics class. After the first examination, a random sample of 6 papers was selected. The grades were 65, 75, 89, 71, 70 and 80. The sample mean is 75 while the sample standard deviation is 8.51. a. Determine the standard deviation of the mean. b. What assumption must be made before we can determine an interval for the mean grade of all the students in the class? Explain why. c. Assume the assumption of part b is met. Provide a 95% confidence interval for the mean grade of all the students in the class.

Solution

a)
Standard deviation( sd )=8.508
b)
Since sigma is estimated from s, we must assume the distribution of all the grades is normal
c)
CI = x ± t a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
ta/2 = t-table value
CI = Confidence Interval
Mean(x)=75
Standard deviation( sd )=8.508
Sample Size(n)=6
Confidence Interval = [ 75 ± t a/2 ( 8.508/ Sqrt ( 6) ) ]
= [ 75 - 2.571 * (3.473) , 75 + 2.571 * (3.473) ]
= [ 66.07,83.93 ]