Question

The researchers suggest that there are occupational differences in mean testosterone level. Medical doctors and university professors are two of the occupational groups for which means and standard deviations are recorded and listed in the following table.

Group	Sample size	Mean	StDev
MD	n1=4n1=4	x¯1=11.17x¯1=11.17	s1=2.07s1=2.07
Prof	n2=3n2=3	x¯2=10.11x¯2=10.11	s2=2.83s2=2.83

Let us denote:

• μ1:μ1: population mean testosterone among medical doctors,
• μ2:μ2: population mean testosterone among university professors,
• σ1:σ1: population standard deviation of testosterone among medical doctors,
• σ2:σ2: population standard deviation of testosterone among university professors.

Case 1: Assume that the population standard deviations are unequal, i.e. σ1≠σ2σ1≠σ2.
1. What is the standard error of the difference in sample mean x¯1−x¯2x¯1−x¯2? i.e. s.e.(x¯1−x¯2)=s.e.(x¯1−x¯2)= [answer to 4 decimal places]

2. Which of the following options gives the formula for 90% confidence interval for μ1−μ2μ1−μ2?
a. 1.06∓2.13×s.e.(x¯1−x¯2)1.06∓2.13×s.e.(x¯1−x¯2)
b. 1.06∓2.78×s.e.(x¯1−x¯2)1.06∓2.78×s.e.(x¯1−x¯2)
c. 1.06∓3.75×s.e.(x¯1−x¯2)1.06∓3.75×s.e.(x¯1−x¯2)
d. 1.06∓1.53×s.e.(x¯1−x¯2)1.06∓1.53×s.e.(x¯1−x¯2)
e. 1.06∓4.6×s.e.(x¯1−x¯2)1.06∓4.6×s.e.(x¯1−x¯2)

3. Are there significant difference between mean testosterone levels of medical doctors and university professors?
a. no, because the 90% confidence interval for μ1−μ2μ1−μ2 contains 0
b. yes, because the entire 90% confidence interval for μ1−μ2μ1−μ2 does not contain 0
c. no, because x¯1=x¯2x¯1=x¯2
d. yes, because x¯1≠x¯2x¯1≠x¯2

Case 2: Now assume that the population standard deviations are equal, i.e. σ1=σ2σ1=σ2.
4. Compute the pooled standard deviation, spooledspooled [answer to 4 decimal places]

5. Which of the following options gives the formula for 90% confidence interval for μ1−μ2μ1−μ2 for pooled situation?
a. 1.06∓2.57×1.83531.06∓2.57×1.8353
b. 1.06∓1.48×1.83531.06∓1.48×1.8353
c. 1.06∓4.03×1.83531.06∓4.03×1.8353
d. 1.06∓3.36×1.83531.06∓3.36×1.8353
e. 1.06∓2.02×1.83531.06∓2.02×1.8353

6. What is the margin of error of the 90% pooled confidence interval of μ1−μ2μ1−μ2? [answer to 4 decimal places]

Answer 1