Question

11. Suppose a random sample of 25 students found a correlation of r = –.53 between number of absences and final grade. Is this enough evidence to conclude that there is a relationship between absences and final grades in general? Use α = .05, two -tailed. Use “Tr” technique as illustrtaed on the PowerPoint lecture. Use of any other formula will be considered incorrect. Include your write up as well. (2 pts) please show all work steps.

Answer 1

We have to test that there is relationship between absences & final grade. We have to use t-test for significance of correlation coefficient.

Hypothesis -

Null hypothesis - H₀ : There is no correlation between absences & final grade, i.e. $\rho$ = 0.

Alternative hypothesis - H₁ : There is correlation between absences & final grade, i.e. $\rho$ $\neq$ 0.

Test statistic -

$t = \frac{r\sqrt{n-2}}{\sqrt{1-r^{2}}}$

Test criterion -

Reject H₀ if t $\geq$ t_{$\alpha$/2,n-2} or -t $\leq$ -t_{$\alpha$/2,n-2} .

Calculations -

We have, r = -0.53, n = 25

So,

$t = \frac{r\sqrt{n-2}}{\sqrt{1-r^{2}}}$

$t = \frac{(-0.53)\sqrt{25-2}}{\sqrt{1-(-0.53)^{2}}}$

$t = \frac{(-0.53)\sqrt{23}}{\sqrt{1-0.2809}}$

$t = \frac{(-0.53)(4.7958)}{\sqrt{0.7191}}$

$t = \frac{-1.8227}{0.8779}$

Critical value -

t_{$\alpha$/2,n-2} = t_0.05/2,25-2 = t_0.025,23 = 2.06866

Conclusion -

-t > -t_{$\alpha$/2,n-2}, so we have to accept H₀ at 5% level of significance.

Result -

There is sufficient evidence that there is no relationship between absences & final grade.

Numbers in each row of the table are values on a t-distribution with (df) degrees of freedom for selected right-tail (greater-than) probabilities (p). tip, df) 0.05 0.0005 0.10 3.077684 1.885618 0.025 12.70620 4.30265 0.005 63.65674 9.92484 636.6192 0.01 31.82052 6.96456 4.54070 3.74695 31.5991 6.313752 2.919986 2.353363 2.131847 الا 1.637744 3.18245 5.84091 12.9240 4.60409 8.6103 2.77645 2.57058 2.015048 3.36493 6.8688 1.943180 1.894579 1.859548 1.833113 1.812461 1.795885 5.9588 5.4079 5.0413 4.7809 4.5869 4.03214 3.70743 3.49948 3.35539 3.24984 3.16927 3.10581 3.05454 3.01228 2.97684 2.94671 2.92078 4.4370 1.782288 43178 4.2208 1.770933 1.761310 1.753050 4.1405 df/p 0.40 0.25 1 0.324920 1.000000 2 0.288675 0.816497 0.276671 0.764892 4 0.270722 0.740697 5 0.267181 0.726687 6 0.264835 0.717558 7 0.263167 0.711142 8 0.261921 0.706387 9 0.260955 0.702722 10 0.260185 0.699812 11 0.259556 0.697445 120.259033 0.695483 13 0.258591 0.693829 14 0.258213 0.692417 15 0.257885 0.691197 16 0.257599 0.690132 17 0.257347 0.689195 18 0.257123 0.688364 19 0.256923 0.687621 20 0.256743 0.686954 21 0.256580 0.686352 22 0.256432 0.685805 23 0.256297 0.685306 24 0.256173 0.684850 25 0.256060 0.684430 26 0.255955 0.684043 27 0.255858 0.683685 28 0.255768 0.683353 29 0.255684 0.683044 30 0.255605 0.682756 Z 0.253347 0.674490 CI 4.0728 1.745884 4.0150 3.9651 2.89823 1.533206 1.475884 1.439756 1.414924 1.396815 1.383029 1.372184 1.363430 1.356217 1.350171 1.345030 1.340606 1.336757 1.333379 1.330391 1.327728 1.325341 1.323188 1.321237 1.319460 1.317836 1.316345 1.314972 1.313703 1.312527 1.311434 1.310415 1.281552 80% 2.87844 3.9216 2.44691 2.36462 2.30600 2.26216 2.22814 2.20099 2.17881 2.16037 2.14479 2.13145 2.11991 2.10982 2.10092 2.09302 2.08596 2.07961 2.07387 2.06866 2.06390 2.05954 2.05553 2.05183 2.04841 2.04523 2.04227 1.95996 95% 1.739607 1.734064 1.729133 1.724718 1.720743 1.717144 1.713872 1.710882 1.708141 1.705618 1.703288 1.701131 1.699127 1.697261 1.644854 3.14267 2.99795 2.89646 2.82144 2.76377 2.71808 2.68100 2.65031 2.62449 2.60248 2.58349 2.56693 2.55238 2.53948 2.52798 2.51765 2.50832 2.49987 2.49216 2.48511 2.47863 2.47266 2.46714 2.46202 2.45726 2.32635 98% 2.86093 2.84534 2.83136 2.81876 2.80734 2.79694 2.78744 2.77871 2.77068 3.8834 3.8495 3.8193 3.7921 3.7676 3.7454 3.7251 3.7066 3.6896 2.76326 3.6739 2.75639 3.6594 3.6460 2.75000 2.57583 3.2905 90% 99% 99.9%