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)
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 |
|
|
|
With df = 10 and confidence level 95% (alpha is 5%), looking at
the value of t as per below table
t value is 1.812
Using below formula to calculate 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
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
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-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 =