Question

A survey cunducted o de errm'าษ whether hours u eep per r "yht dre r dependent of dye. A sample U Individuals was e'sked o indicate he nur er u huurs o sleep per mqht with ca euricd options: ev e u dn 6 hours, 6 to 6.9 hours 7 to 7.9 hours dnd ε huurs or more. Later in the survey, the individuas were asked to indicate their age with categorical options: age 39 or yournger and age 40 r older. Sample data fellow Age Group Hours of Sleep 39 or younger 40 or older 7 to 7.9 73 u more (a) Conduct a teet of independence to determine whether hours of sleep are independent ot age. stote the nul and altenative hypothcses Hú: The prupur Lion of peuple whu yel 3 งเ mure hours uf'sleep per night is not equal acruss Lhe Iwo dge uruups. H The propoition of paapla wha get 8 or mara hours of sleap per night is qual acros tha two aga groups. H: Hours of sleap per night is not indepandent at age. : Hours of sleep per rnight is independent of age. : Hours of sleep per nigh: is mutually exclusive from age : Hours ot sleep per night is not mutually exclusve trom age. Mn Hours of sleep per night is independent of age на: Hours of slecp per night is not independent of agc. Find the value of the test statistic. (Round your answer to three dedimal places.) whet is the p-v? (Round your answer to our decimal pleces.) p-value- uning a a.05 laval af significanca, what is your candusion? 。Do not reject Ho. we conclude that hours of sleep per night and age are independent. Rojoct Ho. We cannot conclude that hours of sleep par night and age are independent. Do not raject Ho.Wa cannot conclude that hours of slasp per night and age ara not Indapandent. ● Reject HO. We conclude that hours uf sleep per night and age dre no, independent. (b) What is your estimate of the percentages of individuals who sleep fewer than hcurs. ō to 0.9 hours, 7 to 7.9 hours, and g hours or more per night? Fewer than 6 to 7.9 อ or more

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

H₀: Hours of sleep per night and age group are independent.
H_a: Hours of sleep per night and age group are not independent.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square test for independence.

Analyze sample data. Applying the chi-square test for independence to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = (r - 1) * (c - 1) = (2 - 1) * (4 - 1)
D.F = 3
E_r,c = (n_r * n_c) / n

	39 or younger	Expected	[(Or,c - Er,c)2/ Er,c]	40 or older	Expected	[(Or,c - Er,c)2/ Er,c]	Total
Fewer than 6	40	37	0.175	38	41	0.1616	78
6 to 6.9	60	56	0.2626	57	61	0.2424	117
7 to 7.9	75	71	0.22074	73	77	0.2038	148
8 or more	65	75	1.4242	92	82	1.315	157
Total	240	240	2.083	260	260	1.9228	500

$\chi ^{2}=\sum \left [\frac{(O_{r,c}-E_{r,c})^{2}}{E_{r,c}} \right ]$
Χ² = 4.01

where DF is the degrees of freedom.

The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 4.01.

We use the Chi-Square Distribution Calculator to find P(Χ² > 16.2) = 0.26

Interpret results. Since the P-value (0.26) is greater than the significance level (0.05), we have to accept the null hypothesis. Thus, we conclude that there is a relationship between Hours of sleep per night and age group.

Do not reject H0, Hours of sleep per night and age group are independent.

b)

	Frequency	Percentage
Fewer than 6	78	15.6%
6 to 6.9	117	23.4%
7 to 7.9	148	29.6%
More than 8	157	47.1%
Total	500	100%