Is my work correct (easy problem, confidence intervals)

Question

The r.v. $X$ represents the time taken by a computer in company $1$ in order to perform a certain job, and $Y$ represents the same thing but for company $2$. A sample of $n_X = 12$ computers are taken from company $1$, and we obtain: $\bar x = 65$, $s_X ^2 = 279$. A sample of $n_Y = 8$ computers are taken from company $2$ and we get $\bar y = 48$, $s_Y ^2 = 224$.

I am required to find a $.95$ confidence interval for the difference between the means of the two populations.

What I did:

Because $\bar x > \bar y$ let's find the C.I for the difference $\mu_X - \mu_Y$. To do this, we note that:

The variances are unknown, and $n_X + n_Y - 2 = 18 \le 30$ is small. Then, we must consider:

$$T = \frac{(\bar X - \bar Y) - (\mu_X - \mu_Y)}{\hat \sigma \sqrt{\frac1{n_X} + \frac1{n_Y}}}$$

Where:

$$\hat \sigma^2 = \frac{n_X S_X ^2 + n_Y S_Y ^2}{n_X + n_Y -2}$$

$T$ has a t-student distribution with degrees of freedom $\nu = n_X + n_Y - 2 = 18$.

$$- t \le T \le t \iff - t \le \frac{(\bar X - \bar Y) - (\mu_X - \mu_Y)}{\hat \sigma \sqrt{\frac1{n_X} + \frac1{n_Y}}} \le t \iff ... \iff \\ (\bar X - \bar Y) - t \hat \sigma \sqrt{\frac1{n_X} + \frac1{n_Y}} \le \mu_X - \mu_Y \le (\bar X - \bar Y) + t \hat \sigma \sqrt{\frac1{n_X} + \frac1{n_Y}}$$

Now we find $t$ from the table, and replace all the known values to get:

C.I $= \left[ 0.457, 33.542 \right]$

I don't care about the part with calculations, but my question is:

Is my work correct?

The next part of the question is asking to find whether we can say the company $1$ has faster computers than company $2$ at a risk $.05$. I know how to do this by testing the hypothesis $\mu_X = \mu_Y$ against $\mu_X > \mu_Y$. But is there a way to do it that makes use of the first part?

Answer 1

Your formulas are correct, but the calculations might not be very accurate.

n1<-12
n2<-8
x_bar<-65
y_bar<-48
sx_2<-279
sy_2<-224

sp<-sqrt(((n1-1)*sx_2+(n2-1)*sy_2)/(n1+n2-2))


t<-qt(0.975,n1+n2-2)#with 0.975 and 18 df


CL1<-(x_bar-y_bar)-t*sp*sqrt(1/n1+1/n2) #lower 95% CI
CL1

#1.608831

CL2<-(x_bar-y_bar)+t*sp*sqrt(1/n1+1/n2) #higher 95% CI
CL2

# 32.39117

95% CI: 1.61-32.4

Since the 95% CI does not include 0, I think you can say company 1 's computer is faster than company 2 at 0.05 type I error level.