In: Statistics and Probability
The number of hours 10 students spent studying for a test and their scores on that test are shown in the table. Is there enough evidence to conclude that there is a significant linear correlation between the data? Use alphaequals0.05.
Hours, x= 0 1 2 4 4 5 5 6 7 8 Test score, y= 41 40 51 56 65 70 76 70 84 97
1. Setup the hypothesis for the test.
2. Identify the critical value(s). Select the correct choice below and fill in any answer boxes within your choice The critical value is ____ Or The critical values are -t0=____ &t0=____
3. Calculate the test statistic. T=_____(Round to three decimal places as needed)
4. What is your conclusion? There ____(is or is not) enough evidence at the 5% level of significance to conclude that there____ (is is not) a significant linear correlation between vehicle weight and variability in braking distance on a dry surface.
X | Y | XY | X² | Y² |
0 | 41 | 0 | 0 | 1681 |
1 | 40 | 40 | 1 | 1600 |
2 | 51 | 102 | 4 | 2601 |
4 | 56 | 224 | 16 | 3136 |
4 | 65 | 260 | 16 | 4225 |
5 | 70 | 350 | 25 | 4900 |
5 | 76 | 380 | 25 | 5776 |
6 | 70 | 420 | 36 | 4900 |
7 | 84 | 588 | 49 | 7056 |
8 | 97 | 776 | 64 | 9409 |
Ʃx = | 42 |
Ʃy = | 650 |
Ʃxy = | 3140 |
Ʃx² = | 236 |
Ʃy² = | 45284 |
Sample size, n = | 10 |
x̅ = Ʃx/n = 42/10 = | 4.2 |
y̅ = Ʃy/n = 650/10 = | 65 |
SSxx = Ʃx² - (Ʃx)²/n = 236 - (42)²/10 = | 59.6 |
SSyy = Ʃy² - (Ʃy)²/n = 45284 - (650)²/10 = | 3034 |
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 3140 - (42)(650)/10 = | 410 |
Slope, b = SSxy/SSxx = 410/59.6 = 6.8791946
y-intercept, a = y̅ -b* x̅ = 65 - (6.87919)*4.2 = 36.107383
Regression equation :
ŷ = 36.1074 + (6.8792) x
Sum of Square error, SSE = SSyy -SSxy²/SSxx = 3034 - (410)²/59.6 = 213.5302013
Standard error, se = √(SSE/(n-2)) = √(213.5302/(10-2)) = 5.16636
--
1.
Null and alternative hypothesis:
Ho: β₁ = 0
Ha: β₁ ≠ 0
2.
df = n-2 = 8
Critical value, t_c = T.INV.2T(0.05, 8) = 2.306
The critical values are -t0 = -2.306 & t0 = 2.306
3.
Test statistic:
t = b/(se/√SSxx) = 10.280
4.
Conclusion:
There is enough evidence at the 5% level of significance to conclude that there is a significant linear correlation between vehicle weight and variability in braking distance on a dry surface.