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 ?1 be the mean time spent lying down by the lean group, and ?2 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 1 506.100 374.300 559.500 Lean 2 602.925 376.512 455.650 Lean 3 315.212 587.138 536.362 Lean 4 587.644 352.144 487.269 Lean 5 582.869 353.994 516.081 Lean 6 538.388 390.312 505.500 Lean 7 672.188 265.188 471.700 Lean 8 552.656 319.219 562.006 Lean 9 373.831 533.031 529.431 Lean 10 500.700 532.838 396.962 Obese 11 264.244 645.281 516.044 Obese 12 463.756 457.644 514.931 Obese 13 362.138 578.662 559.300 Obese 14 412.667 578.662 532.208 Obese 15 346.375 575.662 499.931 Obese 16 418.531 563.556 443.856 Obese 17 363.650 617.262 460.550 Obese 18 269.344 647.181 505.981 Obese 19 414.631 568.769 443.706 Obese 20 426.356 595.369 414.919 (a) What is the practical question that requires a statistical test? A.Do lean and obese people differ in the average time they spend sitting? b.Do lean and obese people differ in the average time they spend lying down? C. Does the average time spent sitting or standing differ from the average time spent lying down for lean and obese people? d.Does the average time spent sitting differ from the average time spent lying down for lean and obese people? (b) State the null and alternative hypotheses. H0: ?1 > ?2 Ha: ?1 ≠ ?2H0: ?1 = ?2 Ha: ?1 > ?2 H0: ?1 ≠ ?2 Ha: ?1 = ?2H0: ?1 = ?2 Ha: ?1 ≠ ?2 (c) Find the size, mean and standard deviation of each group. n x s Lean Obese

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 ?1 be the mean time spent lying down by the lean group, and ?2 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 1 506.100 374.300 559.500 Lean 2 602.925 376.512 455.650 Lean 3 315.212 587.138 536.362 Lean 4 587.644 352.144 487.269 Lean 5 582.869 353.994 516.081 Lean 6 538.388 390.312 505.500 Lean 7 672.188 265.188 471.700 Lean 8 552.656 319.219 562.006 Lean 9 373.831 533.031 529.431 Lean 10 500.700 532.838 396.962 Obese 11 264.244 645.281 516.044 Obese 12 463.756 457.644 514.931 Obese 13 362.138 578.662 559.300 Obese 14 412.667 578.662 532.208 Obese 15 346.375 575.662 499.931 Obese 16 418.531 563.556 443.856 Obese 17 363.650 617.262 460.550 Obese 18 269.344 647.181 505.981 Obese 19 414.631 568.769 443.706 Obese 20 426.356 595.369 414.919 (a) What is the practical question that requires a statistical test? A.Do lean and obese people differ in the average time they spend sitting? b.Do lean and obese people differ in the average time they spend lying down? C. Does the average time spent sitting or standing differ from the average time spent lying down for lean and obese people? d.Does the average time spent sitting differ from the average time spent lying down for lean and obese people? (b) State the null and alternative hypotheses. H0: ?1 > ?2 Ha: ?1 ≠ ?2H0: ?1 = ?2 Ha: ?1 > ?2 H0: ?1 ≠ ?2 Ha: ?1 = ?2H0: ?1 = ?2 Ha: ?1 ≠ ?2 (c) Find the size, mean and standard deviation of each group. n x s Lean Obese

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.6: Summarizing Categorical Data

Problem 23PPS

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Time (minutes per day) spent in three different postures by lean and obese subjects
Group	Subject	Stand/Walk	Sit	Lie
Lean	1	506.100	374.300	559.500
Lean	2	602.925	376.512	455.650
Lean	3	315.212	587.138	536.362
Lean	4	587.644	352.144	487.269
Lean	5	582.869	353.994	516.081
Lean	6	538.388	390.312	505.500
Lean	7	672.188	265.188	471.700
Lean	8	552.656	319.219	562.006
Lean	9	373.831	533.031	529.431
Lean	10	500.700	532.838	396.962
Obese	11	264.244	645.281	516.044
Obese	12	463.756	457.644	514.931
Obese	13	362.138	578.662	559.300
Obese	14	412.667	578.662	532.208
Obese	15	346.375	575.662	499.931
Obese	16	418.531	563.556	443.856
Obese	17	363.650	617.262	460.550
Obese	18	269.344	647.181	505.981
Obese	19	414.631	568.769	443.706
Obese	20	426.356	595.369	414.919

(a)

What is the practical question that requires a statistical test?

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

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

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

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

(b)

State the null and alternative hypotheses.

H₀: ?₁ > ?₂
H_a: ?₁ ≠ ?₂H₀: ?₁ = ?₂
H_a: ?₁ > ?₂ H₀: ?₁ ≠ ?₂
H_a: ?₁ = ?₂H₀: ?₁ = ?₂
H_a: ?₁ ≠ ?₂

(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.

A.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.

b.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.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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