Question

Please use Excel

Exercise (2 p) 1. Research group has developed a new powerline for electric vehicles. Powerline is more efficient and will incrase the operation distance of the car. Let X, be the random variable what describes the increase in operation distance of car i (unit km/kWh). Based on theoretical calculations, the expected value of increased distance is μ-E(X)-1 km/kWh. i) One of the largest car manufacturers in the world investigates the power line in extensive research where the number of the cars is n = 950, As a result of the tests the arithmetic mean of the increased distance is 1,01 and sample standard deviation is s 0,1 Research group decides to make also their own research but because of the lack of resources the number of the cars is only n = 11. This research gives the results of 1,22 and s0,41 ii) Make a statistical test for both researches and present your conclusions based on that. Use risk level of α-5 %. Do the research results support the theoretical calculations?

Answer 1

To solve this, we need to do 2 sample t test

Sample 1 - Extensive research with n = 950

Sample 2 - Own research with n = 11

Running the test (using formulas below)

DF The degrees of freedom are used to determine the T distribution from which T* is generated For the equal variance case df = n1 + n2-2 For the unequal variance case Mean Difference This is the difference between the sample means, X] - X2 Standard Deviation In the equal variance case, this quantity is X1-X2 n1 + n2 - 2 In the unequal variance case, this quantity i s:

Standard Error This is the estimated standard deviation of the distribution of differences between independent sample means For the equal variance case n1 + n2 - 2 For the unequal variance case SEX,-X2= This is the t-value used to construct the confidence limits. It is based on the degrees of freedom and the confidence level Lower and Upper Confidence Limits These are the confidence limits of the confidence interval for μ1-μ2. The confidence interval formula is The equal-variance and unequal-variance assumption formulas differ by the values of T* and the standard error

n1	950		n2	11
µ 1	1.01		µ 2	1.22
s 1	0.1		s 2	0.41
s1 ^ 2	0.01		S2 ^ 2	0.1681
(s1 ^ 2)/n1 + (s2 ^ 2)/n2	0.015292344
((s1 ^ 2)/n1 + (s2 ^ 2)/n2)^2	0.000233856
(s1 ^ 2/n1) ^ 2 / (n1 -1)	1.16758E-13
(s2 ^ 2/n2) ^ 2 / (n2 -1)	2.33534E-05
	2.33534E-05
df	10.01

s (n1-n2)

0.422018957

SE (n1-n2)

0.123662219

With df = 10 and confidence level 95% (alpha is 5%), looking at the value of t as per below table

Table of t-Distribution Areas df 0.25 0.20 0.15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 1 1.000 1.376 1.963 3.078 6.314 12.706 15.894 31.82 63.657127.321 318.309 636.619 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14089 22.327 31.599 0.215 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3 0.765 0.978 1.250 1.638 2.3533.182 3.482 4.541 5.841 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 7.453 10.215 7.173 5.893 5.208 4.785 4.501 4.297 4.144 4.025 3.930 3.852 3.787 3.733 3.686 3.222 3.646 5.598 4.773 4.317 4.029 3.833 3.690 3.581 5 0.727 0.920 1.156 1.476 2.015 2.57 2.757 3.365 4.032 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.7643.169 11 0.697 0.876 1.088 1.363 1.796 2.2012.328 2.718 3.106 12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 13 0.694 0.870 1.079 1.350 77 2.160 2.282 2.650 3.012 3.497 3.428 3.372 14 0.692 0.868 1.076 1.345 1.76 2.145 2.264 2.624 2.977 3.326 3.286 3.252 15 0.691 0.866 1.074 1.34 1.753 2.13 2.249 2.602 2.947 16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 18

t value is 1.812

Using below formula to calculate t statistic

T-Statistio The T-Statistic is the value used to produce the p-value (Prob Level) based on the T distribution. The formula for the T-Statistic is: T - Statistic -Xi-X2- Hypothesized Difference T-Statistic = SEx,-X2 In this instance, the hypothesized difference is zero, so the T-Statistic formula reduces to TStatistic SE T-Statistic =

We get

Two-Sample T-Test and CI

Method

μ₁: mean of Sample 1

µ₂: mean of Sample 2

Difference: μ₁ - µ₂

Equal variances are not assumed for this analysis.

Descriptive Statistics

Sample	N	Mean	StDev	SE Mean
Sample 1	950	1.01	1.22	0.040
Sample 2	11	0.100	0.410	0.12

Estimation for Difference

Difference	95% CI for Difference
0.910	(0.627, 1.193)

Test

Null hypothesis	H₀: μ₁ - µ₂ = 0
Alternative hypothesis	H₁: μ₁ - µ₂ ≠ 0

T-Value	DF	P-Value
7.01	12	0.000

Since p-value is less than the level of significance, we reject the null hypothesis and conclude that there is a difference between the results of two researches