Question

Suppose you are a researcher in a hospital. You are experimenting with a new tranquilizer. You collect data from a random sample of 9 patients. The period of effectiveness of the tranquilizer for each patient (in hours) is as follows:

Hours

2.3

2.8

2.4

2.8

3

3

2.9

2.3

2.9

a) What is a point estimate for the population mean length of time. (Round answer to 4 decimal places)

  

b) What must be true in order to construct a confidence interval in this situation?

  Select an answer The sample size must be greater than 30 The population mean must be known The population must be approximately normal The population standard deviation must be known

c) Construct a 90% confidence interval for the population mean length of time. Enter your answer as an open-interval (i.e., parentheses example (5.2314,8.1245)) Round upper and lower bounds to 4 decimal places.

d) What does it mean to be "90% confident" in this problem?

i) There is a 90% chance that the confidence interval contains the sample mean time.

ii) 90% of all confidence intervals found using this same sampling technique will contain the population mean time.

iii) The confidence interval contains 90% of all sample times.

iv) 90% of all times will fall within this interval.

e) Suppose that the company releases a statement that the mean time for all patients is 2 hours.

Is this possible?  Yes/No

Is it likely? No/Yes

Answer 1

	x	(x-xbar)^2
	2.3	0.169012
	2.8	0.007901
	2.4	0.09679
	2.8	0.007901
	3	0.083457
	3	0.083457
	2.9	0.035679
	2.3	0.169012
	2.9	0.035679
Sum	24.4	0.688889

Mean(x)=xbar=sum(x)/n	2.711111
standard deviation(s)=sum(x-xbar)^2/n-1	0.293447

a)

point estimate for the population mean length of time = 2.7111

b)

The population must be approximately normal

c)

sample standard deviation, s = 0.293
sample size, n = 9
degrees of freedom, df = n - 1 = 8

Given CI level is 98%, hence α = 1 - 0.90 = 0.1

α/2 = 0.1/2 = 0.05, tc = t(α/2, df) =1.8595

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (2.7111 - 1.8595 * 0.393/sqrt(9) , 2.7111+1.8595 * 0.393/sqrt(9))
CI = (2.5292, 2.893)

d)

90% of all times will fall within this interval.

e)

No, it is not possible

No, it is not likely