Answer:
1) The null and alternative hypotheses.
H0: Hours of sleep per night is independent of age.
Ha: Hours of sleep per night is not independent of age.
Test Statistics χ²=4.009
The corresponding p-value for the test at χ²= 4.009 is 0.2604
Step-by-step explanation:
1) The null and alternative hypotheses.
H0: Hours of sleep per night is independent of age.
Ha: Hours of sleep per night is not independent of age.
2) The data is
≤ 39 ≥ 40 Total
6 hours 36 34 70
6.9-7 hours 60 57 117
7- 7.9 hours 79 77 156
<u>8 hours or more 65 92 157</u>
<u>Total 240 260 500</u>
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3) The expected values are computed as follows
260×70/500=36.4
240×117/500 =56.16
260×117/500 =60.84
240×156/500 =74.88
260×156/500 =81.12
240×157/500=75.36
260×157/500 =81.64
4) The squared distances are calculated as
{(36 - 33.6)^2}/{ 33.6} = 0.171
{(34 - 36.4)^2}{ 36.4} = 0.158
{(60 - 56.16)^2}/{ 56.16} = 0.263
{(57 - 60.84)^2}/{ 60.84} = 0.242
{(79 - 74.88)^2}/{ 74.88} = 0.227
{(77 - 81.12)^2}/{ 81.12} = 0.209
{(65 - 75.36)^2}/{ 75.36} = 1.424
{(92 - 81.64)^2}{ 81.64} = 1.315
5) The significance level is set at α=0.05 for df = (4 - 1) (2 - 1) =3
degrees of freedom.
6) The critical region for this test is χ² = χ² >7.815
7) Test Statistics
χ²=i=∑(O ij −Eij )2/Eij
=0.171+0.263+0.227+1.424+0.158+0.242+0.209+1.315
= 4.009
8) Conclusion
As the calculated value χ² =4.009 does not lie in the critical region ≤χ² =7.815, null hypothesis is not rejected.
Therefore, we conclude that there is not enough evidence to accept the alternate hypothesis , at the α=0.05 significance level.
The p-value for the test at χ²= 4.009 is 0.2604
