US mean	US sd	EU & JP mean	EU & JP sd	difference of means	test statistic	p-value
P	6101.38	3088.78	6407.14	2601.61	-305.75	-0.44	0.66
M	19.79	4.68	24.86	6.64	-5.08	-3.26	0.00
H	3.17	0.88	2.55	0.53	0.63	3.77	0.00
R	27.23	2.89	25.84	3.51	1.39	1.64	0.11
Tr	14.75	4.27	11.36	3.24	3.39	3.72	0.00
W	3305.38	710.06	2314.55	434.65	990.84	7.33	0.00
L	196.23	20.44	168.77	13.26	27.46	6.86	0.00
T	41.67	3.69	35.36	1.56	6.31	10.33	0.00
D	233.35	85.89	112.09	24.02	121.26	9.35	0.00
G	2.82	0.35	3.48	0.30	-0.66	-8.15	0.00

Table of sample means and sample standard deviations of observed random variables where we distinguish between American and European & Japanese cars. Last three columns of the table contain value of test statistic of Welch test of equal mean values and its \(p\)-value.

We will assume that the data come from some multivariate normal distribution and we will test the difference between theoretical mean values of vectors \(\mu_{\text{US}}\) and \(\mu_{\text{EU&JP}}\), where \(\mu_{\text{US}}\) denotes the theoretical mean value of vector \[\begin{equation*} \begin{pmatrix} \text{P} \\ \text{R} \end{pmatrix} \end{equation*}\] of American cars and similarly for European and Japanese cars (P - price of the car, R - rear seat clearence of the car).

Testing difference of mean values of other covariates jointly doesnt make sense to me since we already know that the marginal mean values are statistically different (do people say statistically different ?).\ \ Hypothesis and alternative: \[\begin{equation*} H_0 : \mu_{\text{US}} = \mu_{\text{EU&JP}}, \, H_1 : \mu_{\text{US}} \neq \mu_{\text{EU&JP}}, \end{equation*}\] \(p\)-value: \[\begin{equation*} 0.09. \end{equation*}\] Hence we don’t deny the null hypothesis and we can’t say that the difference between mean values of prices and rear seat clearences of American and European & Japanese cars is statisticaly significant – those cars are different in every other observed aspect though.

## Test stat:  4.9622 
## Numerator df:  2 
## Denominator df:  71 
## P-value:  0.09385

Contours for estimated difference.

Confidence intervals for joint data.

##   Lower Boundary Upper Boundary Mean Estimate
## P     5313.99471     7070.57285    6192.28378
## R       25.88281       27.75233      26.81757

Confidence intervals for American cars.

##   Lower Boundary Upper Boundary Mean Estimate
## P      4999.3158     7203.45344    6101.38462
## R        26.2005       28.26103      27.23077

Confidence intervals for European and Japanese cars.

##   Lower Boundary Upper Boundary Mean Estimate
## P     4869.57817     7944.69456    6407.13636
## R       23.76661       27.91521      25.84091