Answer: The 95% confidence interval for the difference between population means μ1-μ2 is [tex](-9.36,\ 15.96)[/tex] .
Step-by-step explanation:
Given : Samples from two independent, normally-distributed populations produced the following results.
Population 1 Population 2
Sample size 7 9
Sample mean 15.9 12.6
Sample standard deviation 10.2 13.4
The confidence interval for the difference between population means μ1-μ2 is given by :-
[tex]\overline{X_1}-\overline{X_2}\pm t^*\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}[/tex]
, where [tex]n_1[/tex] = sample size from population 1
[tex]n_1[/tex] = sample size from population 2
[tex]\overline{X_1}-\overline{X_2}[/tex] = Difference between sample mean of two population
[tex]s_1[/tex]= Sample standard deviation of population 1.
[tex]s_2[/tex]= Sample standard deviation of population 2.
t* = Critical value for [tex]df=n_1+n_2-2[/tex] and significance [tex]\alpha/2[/tex].
As per given :
[tex]n_1=7[/tex] [tex]n_2=9[/tex]
df = 7+9-2=14
[tex]\overline{X_1}-\overline{X_2}=15.9-12.6=3.3[/tex]
[tex]s_1=10.2[/tex] [tex]s_2=13.4[/tex]
[tex]\alpha=1-0.95=0.05[/tex]
Critical t-value : [tex]t_{df, \alpha/2}=t_{14, 0.025}=2.145[/tex]
So , the 95% confidence interval for the difference between population means μ1-μ2 would be
[tex]3.3\pm (2.145)\sqrt{\dfrac{10.2^2}{7}+\dfrac{13.4^2}{9}}[/tex]
[tex]3.3\pm (2.145)\sqrt{14.86+19.95}[/tex]
[tex]3.3\pm (2.145)\sqrt{34.81}[/tex]
[tex]3.3\pm (2.145)(5.9)[/tex]
[tex]3.3\pm 12.66[/tex]
[tex](3.3-12.66,\ 3.3+12.66)=(-9.36,\ 15.96)[/tex]
Hence, the 95% confidence interval for the difference between population means μ1-μ2 : [tex](-9.36,\ 15.96)[/tex]
