Question

QUESTION 3: A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length of time they last is approximately normal. Researchers in a hospital used the drug on a random sample of nine patients. The effective period of the tranquilizer for each patient (in hours) was as follows: 2.7, 2.8, 3.0, 2.3, 2.3, 2.2, 2.8, 2.1 and 2.4.

a) Find the mean of the sample.

b) Find the standard deviation of the sample.

c) Find the median of the sample.

d) Based on the mean and the standard deviation in the sample, estimate the mean amount of time that the tranquilizer is effective in the whole population by constructing and interpreting a 90% confidence interval. (Assume the required conditions are held.) Here you don’t need to show how the interval is built. Just the interval and its interpretation.

Answer 1

	X	X2
	2.7	7.29
	2.8	7.84
	3	9
	2.3	5.29
	2.3	5.29
	2.2	4.84
	2.8	7.84
	2.1	4.41
	2.4	5.76
total	22.6	57.56

a)

Let X be the number of hours the effect of tranquilizer last for all patients.

Here,

n = 9

Therefore, the mean of the sample is 2.511

b)

The standard deviation is given by,

Therefore, the standard deviation of the sample is 0.3180

c)

For finding median, arrange the data in ascending order and find the middle value using the formula.

The data in ascending order is given below:

X
2.1
2.2
2.3
2.3
2.4
2.7
2.8
2.8
3

Here, n = 9 ( odd)

The formula for finding median for odd number of observations is,

So,

= 2.4

Therefore, the median of the sample is 2.4

d)

Let X be the number of hours the effect of tranquilizer last for all patients.   

The distribution needed for this problem is t- distribution with (n−1) degrees of freedom because we do not know the population standard deviation.

Degrees of freedom = n -1 = 9 -1 = 8

From t-table, the critical t-value for 90% confidence level with 8 degrees of freedom is 1.860

i.e.

Now

  

( 2.3138 , 2.7082)

Therefore, the 90% confidence interval for population mean is ( 2.3138 , 2.7082)

With 90% confidence, it can be said that the population mean length of the time for the effect of
tranquilizer lies between 2.3138 hours and 2.7082 hours.