Question

Going back to problem 1, in real life you can, without much difficulty, get the mean grades of Prof. Lax’s classes but that is about it; meaning you will have no idea how his grades would be distributed, nor would you have any idea about the standard deviation of these grades. (I doubt Prof. Lax would advertise his laxness on his website. Contrary what you might believe that is academically bad form and might negatively affect his students’ hireability in the job market). However, you have access to Miss Z’s data (which she swears is obtained by a random selection process) and the grades she obtained in her random sample of nine were:

79, 75, 84, 63, 98, 52, 87, 99, 83

a .to help Miss Z with her decision to take this course with Prof. Lax or not, create a 97% confidence interval (CI) for the mean using Miss Z.’s data. Make sure that you do the necessary checks.

b. Does your interval capture the rumored population mean of 85?

c. Calculate the margin of error (ME or simply E) of your confidence interval.

d. Miss Z thinks a margin of error (or E) of 7 points or more will have a significant negative effect on her GPA. How does the ME (or E) of your 97% CI from part (c) compare to what she says her GPA can afford? If your CI’s ME (or E) is different than 7 points she can afford what are the ways you can use to reduce the margin of error down to 7 or smaller. Discuss all that can be done. 3

Answer 1

a) The 97% confidence interval is

where

= (79+75+......+83) /9

= 80

= 15.24

	x	(x-xbar)^2
	79	1
	75	25
	84	16
	63	289
	98	324
	52	784
	87	49
	99	361
	83	9
sum	720	1858
mean	80
s^2		232.25
s		15.23975

n=9 , df = 8

For 97% confidence , tc = 2.634

Therefore 97% CI is

= (66.62 , 93.38 )

Note : as sample size is small and population standard deviation is not known , we use t test . And for small sample size , we can use t distribution if the population is normally distributed.

b) Yes the confidence interval includes 85 ,

c) ME =

  

= 13.38

d) Margin of error is more than 7

To reduce margin of error ,we can increase the sample size (with decrease of sample size denominator decreases , also with increase of sample size , df increases and with increase of df, tc decreases)

To reduce margin of error , we can decrease the confidence level ( with decrease of confidence level , tc decreases )