Question

People gain weight when they take in more energy from food than they expend. Researchers wanted to investigate the link between obesity and energy spent on daily activity. Choose 20 healthy volunteers who don't exercise. Deliberately choose 10 who are lean and 10 who are mildly obese but still healthy. Attach sensors that monitor the subjects' every move for 10 days. The table below presents data on the time (in minutes per day) that the subjects spent standing or walking, sitting, and lying down. Is there a significant difference between the mean times the two groups spend lying down? Let μ1 be the mean time spent lying down by the lean group, and μ2 be the mean time for the obese group.

Time (minutes per day) spent in three different postures by lean and obese subjects
Group	Subject	Stand/Walk	Sit	Lie
Lean	1	510.100	367.300	553.500
Lean	2	602.925	379.512	452.650
Lean	3	324.212	585.138	534.362
Lean	4	582.644	361.144	494.269
Lean	5	583.869	346.994	512.081
Lean	6	545.388	382.312	502.500
Lean	7	682.188	269.188	472.700
Lean	8	554.656	318.219	565.006
Lean	9	374.831	541.031	531.431
Lean	10	503.700	524.838	392.962
Obese	11	265.244	645.281	522.044
Obese	12	467.756	461.644	517.931
Obese	13	362.138	577.662	564.300
Obese	14	416.667	574.662	537.208
Obese	15	350.375	579.662	500.931
Obese	16	417.531	564.556	449.856
Obese	17	363.650	618.262	462.550
Obese	18	265.344	646.181	506.981
Obese	19	411.631	567.769	453.706
Obese	20	422.356	590.369	414.919

a) What is the practical question that requires a statistical test?

Do lean and obese people differ in the average time they spend lying down?

Does the average time spent sitting differ from the average time spent lying down for lean and obese people?  

   Does the average time spent sitting or standing differ from the average time spent lying down for lean and obese people?

Do lean and obese people differ in the average time they spend sitting?

(b) State the null and alternative hypotheses.

H₀: μ₁ ≠ μ₂
H_a: μ₁ = μ₂H₀: μ₁ = μ₂
H_a: μ₁ ≠ μ₂     H₀: μ₁ = μ₂
H_a: μ₁ > μ₂H₀: μ₁ > μ₂
H_a: μ₁ ≠ μ₂

(c) Find the size, mean and standard deviation of each group.

	n	x	s
Lean
Obese

(d) Calculate the test statistic.
t =  


(e) Describe your results in this setting.

There is not enough evidence at the 5% significance level to reject the hypothesis that lean and moderately obese people spend (on average) the same amount of time lying down.

There is enough evidence at the 5% significance level to reject the hypothesis that lean and moderately obese people spend (on average) the same amount of time lying down.    

Answer 1

Solution

Part (a)

The practical question that requires a statistical test would be:

Do lean and obese people differ in the average time they spend lying down?

Other activities involve expending energy much more. Thus, a difference in time spent on lying down would throw more light on the relation between obesity and expended energy by daily activities. ANSWER

Part (b)

Null and alternative hypotheses.

H₀: μ₁ = μ₂ Vs H_a: μ₁ ≠ μ₂  ANSWER

Part (c)

Size, mean and standard deviation of each group.

	n	xbar	s
Lean	10	501.1461	51.5030
Obese	10	493.0426	46.1769

Part (d)

Test statistic.
t = (Xbar - Ybar)/{s√(2/n)} where

s² = (s₁² + s₂²)/2;

Xbar = sample average for lean sample subjects,

Ybar = sample average for obese sample subjects,

s₁, s₂ are sample standard deviations for lean sample subjects and lean obese subjects respectively, and

n = common sample size (10)

So, t = 0.0305 ANSWER

Details of calculations

n =	10
Xbar =	501.1461
Ybar =	493.0426
s1 =	51.50301
s2 =	46.17691
s^2 =	2392.434
s =	48.91251
tcal =	0.370457
α =	0.05
tcrit =	2.100922
p-value =	0.715366

[tcrit = upper 2.5% point of t-distribution with 18 degrees of freedom and p-value = P(t₁₈ > |0.3705|]

Part (e) Results (conclusion)

[Decision

Since | tcal | < tcrit, H₀ is accepted or equivalently,since p-value > α (0.05 given), H₀ is accepted.]

There is not enough evidence at the 5% significance level to reject the hypothesis that lean and moderately obese people spend (on average) the same amount of time lying down. ANSWER

DONE