Question

Before treatment with a new drug 11 people with sleep problems have a median sleeping time of 2 hours per night. A drug is administered and it is known that if it has an effect it will increase sleeping time, but some doctors doubt if it will have any effect. Are their doubts justified if the hours per night slept by these individuals after taking the drug are: 3.1 1.8 2.7 2.4 2.9 0.2 3.7 5.1 8.32.1 2.4

Answer 1

We will conduct one sample median test which checks whether or not there is a significant difference between the hypothesized median and the real median of a sample.

Null hypothesis H0: Median sleeping time = 2 hours per night

Alternative hypothesis H0: Median sleeping time > 2 hours per night

Step 1: Assign a value of 0 to those with sleeping time below hypothesized median, and 1, to those above. If we have any sleeping time equal to the hypothesized value, they would not be counted at all.

3.1 - 1

1.8 - 0

2.7 - 1

2.4 - 1

2.9 - 1

0.2 - 0

3.7 - 1

5.1 - 1

8.3 - 1

2.1 - 1

2.4 - 1

Step 2: Find the obtained frequency (of). There are 2 people with sleeping time below the median (the median is 2), and 9 above.
of = 2 and 9.

Step 3: Find the expected frequency (ef). The total number of data points—people—is 11. If 2 was a true median, we’d expect to have 5.5 people with sleeping time over 2 and 5.5 below.

ef = 5.5

Step 4: Calculate the chi-square:

Χ² = Σ [(of – ef)²/ef] = (2 – 5.5)²/5.5 + (9 – 5.5)²/5.5 = 4.45

Step 5: Find the degrees of freedom. There are two observed frequencies (equal to two cells in a contingency table), so there is just one degree of freedom.

Step 6: Use a Chi-squared table to find the critical chi-square value
for 1 degree of freedom and an alpha level of 0.05 ($\alpha$ = 0.05). This equals 3.84.

Since our Χ² value of 4.45 is greater than the critical Χ² value of 3.84, we reject the null hypothesis and conclude that there is significant evidence that the median sleeping time is greater than 2. The doubts cannot be justified at significance level of 0.05.