Question

HW 30 #3 Random samples of resting heart rates are taken from two groups. Population 1 exercises regularly, and Population 2 does not. The data from these two samples is given below: Population 1: 71, 63, 65, 70, 71, 67, 70

Population 2: 73, 71, 78, 73, 73, 77, 70, 69

Is there evidence, at an α=0.06 α = 0.06 level of significance, to conclude that there those who exercise regularly have lower resting heart rates? Carry out an appropriate hypothesis test, filling in the information requested.

A. The value of the standardized test statistic:

Note: For the next part, your answer should use interval notation. An answer of the form (−∞,a) ( − ∞ , a ) is expressed (-infty, a), an answer of the form (b,∞) ( b , ∞ ) is expressed (b, infty), and an answer of the form (−∞,a)∪(b,∞) ( − ∞ , a ) ∪ ( b , ∞ ) is expressed (-infty, a)U(b, infty).

B. The rejection region for the standardized test statistic:

C. The p-value is

Answer 1

From the given data: The following have been calculated

	Population 1	Population 2
Total	477	584
n	7	8
Mean	68.14	73
SD	3.185	3.162

Since s₁/s₂ = 3.185 / 3.162 = 1.007 (it lies between 0.5 and 2) we used the pooled standard deviation

The degrees of freedom using pooled variance = n1 + n2 – 2 = 7 + 8 - 2 = 13

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The Hypothesis:

H0:

Ha:

This is a Left tailed test.

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The Test Statistic:

t observed = -2.96

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The t critical value at = 0.06 is -1.664

Therefore the rejection region is (infty, -1.664)

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The p Value:    The p value (Left tail) for t = -2.96, df = 13, is; p value = 0.0055

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Calculation for the mean and standard deviation:

Mean = Sum of observation / Total Observations

Standard deviation = SQRT(Variance)

Variance = Sum Of Squares (SS) / n - 1, where SS = SUM(X - Mean)².

Population 1					Population 2
#	X	X - Mean	(X - Mean)2		#	X	X - Mean	(X - Mean)2
1	71	68.14	8.1796		1	73	73	0
2	63	68.14	26.4196		2	71	73	4
3	65	68.14	9.8596		3	78	73	25
4	70	68.14	3.4596		4	73	73	0
5	71	68.14	8.1796		5	73	73	0
6	67	68.14	1.2996		6	77	73	16
7	70	68.14	3.4596		7	70	73	9
					8	69	73	16
Total	477	SS	60.857		Total	584	SS	70

	Population 1	Population 2
Total	477	584
n	7	8
Mean	68.14	73
SS	60.857	70
Variance	10.1429	10.0000
SD	3.185	3.162