Question

A random sample of 18 female neurologists and 21 male neurologists at public medical schools found that the average salary for female neurologists was $175,000 with a standard deviation of $15,000, while the average salary for male neurologists was $203,000 with a standard deviation of $22,000. Find a 99% confidence interval for the difference between the average salaries of male and female neurologists. (Round your answer to the nearest dollar.)
a) Write the formula that should be used to compute the interval.
b) Write the assumptions that need to be met in order to use the above formula.
c) Compute the interval.

Answer 1

Answer:

a) = x1`-x2`/ Sp √ ( 1/n₁ + 1/n₂)

b)  it is assumed that the population standard deviations are known (which is sometimes met and sometimes not known).

c) The Confidence interval = [ 19131.77, 36868.23]

Step-by-step explanation:

Part a:

The 99% confidence interval for the difference between the average salaries of male and female neurologists is given by

(x1`-x2`) ±t∝/2(υ)* √ ( Sp²/n₁ + Sp²/n₂)

= x1`-x2`/ Sp √ ( 1/n₁ + 1/n₂)

where Sp ²= s₁² ( n1-1) + s₂²(n2-1) /(n1-1) + (n2-1)

Part B:

1) the sample should be normally distributed

2) it is assumed that the population standard deviations are known (which is sometimes met and sometimes not known).

Part C:

Calculations

1) H0: μ₁=μ₂ against the claim Ha: μ₁ ≠μ₂

2)The test statistic is

t= x1`-x2`/ √ ( Sp²/n₁ + Sp²/n₂)

where

Sp ²= s₁² ( n1-1) + s₂²(n2-1) /(n1-1) + (n2-1)

= 15,000²(17) + 22,000²(22) / 37

Sp ²= 103378640

3) D.f = n1+n2-2 =21+18-2 = 37

4) for 2 tailed test the critical value of t is obtained by

t≥ t∝/2 ( d.f)=t≥ t0.005 (37)= ±2.7154= ± 2.72

5) Putting the values

(x1`-x2`) ±t∝/2(υ)* √ ( sp²/n₁ + sp²/n₂)

28000 ± (2.72 * 3265.89)

=[ 19131.77, 36868.23].

The Confidence interval = [ 19131.77, 36868.23]