Two types of flares are tested for their burning times (in minutes) and sample results are given below.
Brand A: $n = 35 , \bar { Y } = 19.4 , s = 1.4$
Brand B: $n = 40 , \bar { Y } = 15.1 , s = 0.8$
Construct a $95 \%$ confidence interval for the difference $\mu _ { A } - \mu _ { B }$ based on the sample data.
A) $( 3.8,4.8 )$
B) $( 3.6,5.0 )$
C) $( 3.2,5.4 )$
D) $( 3.5,5.1 )$
E) $( - 4.7 , - 3.9 )$

Question

Two types of flares are tested for their burning times (in minutes) and sample results are given below. 
Brand A: $n = 35 , \bar { Y } = 19.4 , s = 1.4$
Brand B: $n = 40 , \bar { Y } = 15.1 , s = 0.8$
Construct a $95 \%$ confidence interval for the difference $\mu _ { A } - \mu _ { B }$ based on the sample data.
A) $( 3.8,4.8 )$
B) $( 3.6,5.0 )$
C) $( 3.2,5.4 )$
D) $( 3.5,5.1 )$
E) $( - 4.7 , - 3.9 )$

Quizplus · Accepted Answer

The correct answer is A because the 95% confidence interval for the difference $\mu _ { A } - \mu _ { B }$ is $( 3.8,4.8 )$. This can be calculated using the formula:$$\bar{Y}_A - \bar{Y}_B \pm t_{\alpha/2, n_A+n_B-2} \sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}$$where $\bar{Y}_A$ and $\bar{Y}_B$ are the sample means, $s_A$ and $s_B$ are the sample standard deviations, $n_A$ and $n_B$ are the sample sizes, and $t_{\alpha/2, n_A+n_B-2}$ is the critical value from the t-distribution with $n_A+n_B-2$ degrees of freedom and a significance level of $\alpha/2$.Plugging in the values from the question, we get:$$19.4 - 15.1 \pm 2.024 \sqrt{\frac{1.4^2}{35} + \frac{0.8^2}{40}} = ( 3.8,4.8 )$$

Two Types of Flares Are Tested for Their Burning Times n=35,Yˉ=19.4,s=1.4n = 35 , \bar { Y } = 19.4 , s = 1.4n=35,Yˉ=19.4,s=1.4

Two Types of Flares Are Tested for Their Burning Times $n = 35 , \bar { Y } = 19.4 , s = 1.4$