Question

 

For each problem, perform the following steps. Assume that all variables are normally or approximately normally distributed.

1. State the hypothesis and identify the claim.

2. Find the critical value(s).

3. Compute the test value.

4. Make the decision.

5. Summarize the results.

2. The average temperatures for a 25-day period for Birmingham, Alabama, and Chicago, Illinois, are shown. Based on the samples, at α = 0.10, can it be concluded that it is warmer in Birmingham? [4]

Birmingham					Chicago
78	82	68	67	68	70	74	73	60	77
75	73	75	64	68	71	72	71	74	76
62	73	77	78	79	71	80	65	70	83
74	72	73	78	68	67	76	75	62	65
73	79	82	71	66	66	65	77	66	64

Answer 1

Let be the the average temperature in Birmingham, Alabama.

Let be the the average.in Chicago, Illinois.

From the data, the following was calculated.

For: = 72.92, s₁ = 5.5, n₁ = 25

For: = 72.8, s₂ = 5.82, n₂ = 25

Since s₁/s₂ = 5.5 / 5.82 = 0.95 (it lies between 0.5 and 2) we used the pooled variance.

The degrees of freedom used is n1 + n2 - 2 = 25 + 25 - 2 = 48 (since pooled variance is used)

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The Hypothesis:

H0: : The temperatures in the 2 cities are the same.

Ha: : Birmingham is warmer than Illinois. (Claim)

This is a Right tailed test.

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The Critical Value:   The critical value (Right tail) at = 0.10, df = 48, tcritical = + 1.2994

Therefore Reject H0, If t observed is > 1.2994

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The Test Statistic:

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The Decision:    Since t observed (1.32) is > t critical (1.299), We Reject H0.

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The p Value:    The p value (Right Tail) for t = 1.32, df = 48, is; p value = 0.0959

_______________________________________________________

The Conclusion: There is sufficient evidence at the 90% significance level to conclude that Birmingham, Alabama is warmer than Chicago, Illinois

_______________________________________________________

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.

	Birmingham					Chicago
#	X	Mean	(X-Mean)2		#	X	Mean	(X-Mean)2
1	78	72.92	25.8064		1	70	70.8	0.64
2	75	72.92	4.3264		2	71	70.8	0.04
3	62	72.92	119.2464		3	71	70.8	0.04
4	74	72.92	1.1664		4	67	70.8	14.44
5	73	72.92	0.0064		5	66	70.8	23.04
6	82	72.92	82.4464		6	74	70.8	10.24
7	73	72.92	0.0064		7	72	70.8	1.44
8	73	72.92	0.0064		8	80	70.8	84.64
9	72	72.92	0.8464		9	76	70.8	27.04
10	79	72.92	36.9664		10	65	70.8	33.64
11	68	72.92	24.2064		11	73	70.8	4.84
12	75	72.92	4.3264		12	71	70.8	0.04
13	77	72.92	16.6464		13	65	70.8	33.64
14	73	72.92	0.0064		14	75	70.8	17.64
15	82	72.92	82.4464		15	77	70.8	38.44
16	67	72.92	35.0464		16	60	70.8	116.64
17	64	72.92	79.5664		17	74	70.8	10.24
18	78	72.92	25.8064		18	70	70.8	0.64
19	78	72.92	25.8064		19	62	70.8	77.44
20	71	72.92	3.6864		20	66	70.8	23.04
21	68	72.92	24.2064		21	77	70.8	38.44
22	68	72.92	24.2064		22	76	70.8	27.04
23	79	72.92	36.9664		23	83	70.8	148.84
24	68	72.92	24.2064		24	65	70.8	33.64
25	66	72.92	47.8864		25	64	70.8	46.24

	Birmingham	Chicago
n	25	25
Sum	1823	1770
Mean	72.92	70.80
SS	725.84	812
Variance	30.24	33.83
SD	5.50	5.82