To estimate the difference in the average household income between community A and community B, we can use a confidence interval. A confidence interval is a range of values that we are fairly certain contains the true population parameter.
In this case, we want to estimate the difference in the average household income between the two communities. To do this, we can use the two-sample t-test. The formula for the confidence interval for the difference in means is:
CI = (�X1 - �X2) ± (t * SE)
Where:
- �X1 is the mean household income in community A
- �X2 is the mean household income in community B
- t is the critical value from the t-distribution based on the desired confidence level (in this case, 95% confidence)
- SE is the standard error of the difference in means
First, let's calculate the standard error of the difference in means:
SE = √((s1^2/n1) + (s2^2/n2))
Where:
- s1 is the standard deviation of the household income in community A
- s2 is the standard deviation of the household income in community B
- n1 is the sample size of community A
- n2 is the sample size of community B
Plugging in the values from the problem:
- s1 = Rs. 2,200
- s2 = Rs. 2,800
- n1 = n2 = 50
SE = √((2200^2/50) + (2800^2/50))
Now, let's calculate the t-critical value for a 95% confidence level. The degrees of freedom for this test is (n1 + n2 - 2):
df = (50 + 50 - 2) = 98
Using a t-table or a statistical software, we can find that the critical value for a 95% confidence level and 98 degrees of freedom is approximately 1.984.
Finally, we can calculate the confidence interval:
CI = (44600 - 43800) ± (1.984 * SE)
Plug in the values of SE and calculate the confidence interval.
The resulting confidence interval will give you the estimated range within which the true difference in the average household income between the two communities is likely to fall, with 95% confidence.