Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent simple random samples selected from normally distributed populations. Assume that the population standard deviations are equal.

See the attachment for formatted formulas

Step-by-step explanation:

Let X11, X12, ……,X1n  and X21 , X22……., X2n be two small independent random samples from two normal populations with means u1 and u2  and the standard deviations σ1 and σ2 respectively. If σ1= σ2 (=σ) but unknown then the unbiased pooled or combined estimate of the common variance σ2 (the term variance means that each population has the same variance) is given by

Sp2  = ((n_1-1)  s_(1^2 )+ (n_2-1) s_2^2)/(n_1+n_2-2)

Where

S12 = 1/(n_1- 1) ∑▒〖 (X_1i- X`_1)〗^2  and

S22 = 1/(n_2- 1) ∑▒〖 (X_2j- X`_2)〗^2

The test statistic

t = ((X_1`-X_2`)- (μ_1- μ_2))/(√(s_p&1/n_1 )+ 1/n_2 )

Has t distribution with v= n1 + n2 – 2 degrees of freedom.

It is used as a test statistic for testing hypotheses about the difference between two population means.

The procedure for testing hypothesis H0: μ_1- μ_2= ∆_0 in case of small independent samples when σ_1= σ_2 is as follows.

H0: μ_1- μ_2= ∆_0  against the appropriate alternative.

t = ((X_1`-X_2`)- ∆_0  )/(√(s_p&1/n_1 )+ 1/n_2 )

Which has t distribution with v= n1 + n2 – 2 degrees of freedom.

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