The research investigator is interested in studying whether there is a significant difference in the salaries of MBA grades into metropolitan cities .A random sample of size 100 from Mumbai is yields on average income of Rs 20,150 another random sample of 60 from Chennai results in an average income of Rs 20,250 . if the variances of both the populations are given as σ12 = Rs 40,000 and σ22 = Rs 32,400 respectively
To determine if there is a significant difference in the salaries of MBA graduates in metropolitan cities, we can conduct a two-sample t-test.
First, we will set up the null and alternative hypotheses:
Null Hypothesis (H0): There is no significant difference in the average salaries of MBA graduates in Mumbai and Chennai.
Alternative Hypothesis (H1): There is a significant difference in the average salaries of MBA graduates in Mumbai and Chennai.
Next, we will calculate the t-statistic using the formula:
t = (X̄1 - X̄2) / √[(s1^2/n1) + (s2^2/n2)]
where:
X̄1 = average income of MBA graduates in Mumbai = Rs 20,150
X̄2 = average income of MBA graduates in Chennai = Rs 20,250
s1 = standard deviation of Mumbai population = Rs √40,000 = Rs 200
s2 = standard deviation of Chennai population = Rs √32,400 = Rs 180
n1 = sample size from Mumbai = 100
n2 = sample size from Chennai = 60
Plugging in the values:
t = (20,150 - 20,250) / √[(200^2/100) + (180^2/60)]
t = -100 / √[4000 + 3240]
t = -100 / √(7240)
t = -100 / 85.1
t ≈ -1.175
Now, we will determine the degrees of freedom (df) using the formula:
df = (s1^2/n1 + s2^2/n2)^2 / [(s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1)]
Plugging in the values:
df = [(200^2/100 + 180^2/60)^2] / [(200^2/100)^2/99 + (180^2/60)^2/59]
df = [(4000 + 3240)^2] / [(4000)^2/99 + (3240)^2/59]
df = (7240^2) / (160,000/99 + 104,976/59)
df = 52,137,600 / 1,616.16 + 1,779.93
df ≈ 32,555
Using a t-table or statistical software, we can find the critical t-value for a significance level (α) of, for example, 0.05 with 32,555 degrees of freedom. Our calculated t-value of -1.175 does not exceed the critical t-value, so we fail to reject the null hypothesis. This means that there is no significant difference in the average salaries of MBA graduates in Mumbai and Chennai.