Question

Y ou have been asked by a men's clothing manufacturer, Nuke, to provide advice on whether there is evidence of differences in the average heights of male citizens in three different countries. You have taken random samples of the heights of males (in cms) in those three countries and your data set is attached in csv delimited format (as a .txt file) using the link on the right.

Input the data set into SPSS and perform the appropriate analysis to answer the question above (marked in bold). Follow SPSS instructions to set up the data in the correct format. (You will need to create a new quantitative variable - with four decimals - with all the heights and a new quantitative variable with values 1, 2 or 3 depending on whether the men are from country 1, 2 or 3. Both of those variables have to be defined as "numeric" in SPSS.)

Choose the correct answer to the following questions, based on your results:

a) Based on the outputs from SPSS, and looking at the appropriate table, can the variances be pooled based on this data set?

Yes, because 1.503 is smaller than 2.306

Yes, because 2.226 is smaller than 2.306

Yes, because 1.503 is smaller than 2.226

No, because all standard deviations for the data set are different

b) Possible null and alternative hypothesis for the ANOVA test could include:

i)Ho: X₁=X₂=X₃, where 1,2 and 3 refer to country 1, country 2 and country 3.
Ha: not all sample means are the same.

ii)Ho: μ₁=μ₂=μ₃, where 1,2 and 3 refer to country 1, country 2 and country 3.
Ha: not all sample means are the same.

iii)Ho: μ₁=μ₂=μ₃, where 1,2 and 3 refer to country 1, country 2 and country 3.
Ha: not all population means are the same.

iv)Ho: μ₁=μ₂=μ₃, where 1,2 and 3 refer to country 1, country 2 and country 3.
Ha: all population means are different.

The correct null and alternative hypothesis for the ANOVA test are: iiiiiiiv

c) Give the value of the test statistic to three decimal places:

d) Based on the ANOVA test, at the 1% significance level, we can rejectfail to rejectaccept the null hypothesis.

e) Based on the ANOVA test, at the 1% significance level, we can conclude:

That the average heights of adult males in the three countries are all the same.

That the average heights of adult males in the three countries are not all the same.

That the average heights of adult males in the three countries are all different.

f) At the 1% significance level:

The average height of adult males in country 1 is significantly different from the average height of adult males in country 2.

The average height of adult males in country 1 is significantly different from the average height of adult males in country 3.

The average height of adult males in country 1 is significantly different from the average height of adult males in country 2 and in country 3.

Data:

Country1,Country2,Country3
177.0035,176.899,173.7444
174.2969,178.5841,170.715
176.0359,173.2635,174.3633
175.65,177.3208,171.5132
176.7979,176.6144,170.1127
176.8356,174.5,171.1957
176.8271,175.6363,173.2149
175.8271,176.6454,173.3727
175.5209,180.8519,172.2083
175.5335,179.8003,172.3736
177.4096,174.4451,172.8228
173.8911,180.1454,170.1032
176.0305,177.143,171.2438
176.6515,176.118,169.6208
175.4514,177.1292,171.4162
176.9204,175.9335,173.2341
176.2073,176.0386,172.9327
175.0802,178.1366,174.9742
176.5098,178.0317,172.2167
175.9514,178.0424,169.1966
175.7672,177.0729,174.1962
175.3753,174.6303,173.667
176.3799,177.1324,171.2426
173.9,178.3193,172.2894
175.7822,176.8356,173.0133
175.5934,177.5451,172.5001
175.1145,177.145,172.9053
177.1485,175.8055,171.8535
175.3454,176.582,171.9495
177.0028,175.1765,172.1145
177.5686,177.3549,170.9734
176.0961,174.7088,173.8498
176.6611,174.8105,172.0082
177.1314,175.1477,172.8924
176.1947,172.3724,170.8886
177.0031,178.0699,170.6022
174.6587,176.6227,171.8817
177.7436,175.2186,171.6714
175.0136,177.9814,173.5533
173.9225,173.975,172.7879
178.0998,176.0671,172.7027
176.2185,175.8861,170.1216
176.8836,176.615,170.742
175.8278,176.6067,172.2627
174.9732,175.0757,171.5478
177.1142,176.1609,173.7435
175.9273,175.9857,171.245
176.0435,177.016,173.1377
176.8273,177.6212,173.8141
174.4548,177.6421,172.4173
175.3517,175.0773,174.1104
177.7915,176.3006,173.3812
177.0772,174.6216,172.2783
174.6343,174.7524,173.1398
173.4358,176.1911,172.2527
175.5574,178.1924,171.4606
176.5782,175.1994,172.466
177.5251,176.6828,172.0471
176.4218,175.9067,171.526
174.841,177.6526,174.3023
176.698,174.7842,172.2877
174.776,176.2423,175.5495
175.0507,176.9183,171.615
175.4501,177.6308,172.7336
174.7018,178.2075,174.0574
175.9477,176.3117,175.548
175.1301,174.2609,172.3946
175.9286,175.235,170.9727
175.6989,174.8199,173.0444
175.7673,179.2556,170.2841
175.6454,175.3997,171.644
177.9033,177.1725,172.3534
177.0934,175.9499,173.9364
176.3768,177.3552,173.8683
174.393,175.2057,171.9642
174.1865,174.3771,173.2011
173.7458,174.3509,171.5823
177.5641,176.8347,174.4899
176.6341,175.9694,172.2038
175.0335,175.9451,171.6093
174.8097,178.0451,171.35
177.0119,176.5791,172.9314
178.3346,176.4572,173.7385
174.0987,178.264,174.7634
175.7158,175.1542,172.0703
176.1904,177.1056,172.2119
173.7104,177.2856,171.1775
177.3936,175.8832,174.0956
175.7609,176.4804,171.9019
178.7858,174.6844,171.3848
175.7918,174.7077,174.8395
176.8319,176.3363,173.0528
174.8148,177.1389,172.9312
175.4877,179.5611,170.6928
174.7807,175.333,172.5642
176.8103,176.4435,172.6762
177.9924,176.0928,170.5801
175.5283,173.6871,171.7065

Answer 1

Solution:

Descriptives
height
	N	Mean	Std. Deviation	Std. Error	95% Confidence Interval for Mean		Minimum	Maximum
					Lower Bound	Upper Bound
1.00	98	175.9693	1.15286	.11646	175.7382	176.2004	173.44	178.79
2.00	98	176.3929	1.50329	.15186	176.0915	176.6943	172.37	180.85
3.00	98	172.4094	1.30403	.13173	172.1480	172.6709	169.20	175.55
Total	294	174.9239	2.22558	.12980	174.6684	175.1793	169.20	180.85

ANOVA
height
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	938.205	2	469.103	266.058	.000
Within Groups	513.079	291	1.763
Total	1451.284	293

Test of Homogeneity of Variances
height
Levene Statistic	df1	df2	Sig.
2.204	2	291	.112

a) Based on the outputs from SPSS, and looking at the appropriate table, can the variances be pooled based on this data set?

A)Yes, because 1.503 is smaller than 2.306

Answer: B)Yes, because 2.226 is smaller than 2.306

C)Yes, because 1.503 is smaller than 2.226

D)No, because all standard deviations for the data set are different

b) Possible null and alternative hypothesis for the ANOVA test could include:

i)Ho: X₁=X₂=X₃, where 1,2 and 3 refer to country 1, country 2 and country 3.
Ha: not all sample means are the same.

ii)Ho: μ₁=μ₂=μ₃, where 1,2 and 3 refer to country 1, country 2 and country 3.
Ha: not all sample means are the same.

iii)Ho: μ₁=μ₂=μ₃, where 1,2 and 3 refer to country 1, country 2 and country 3.
Ha: not all population means are the same.

iv)Ho: μ₁=μ₂=μ₃, where 1,2 and 3 refer to country 1, country 2 and country 3.
Ha: all population means are different.

The correct null and alternative hypothesis for the ANOVA test are: ????

Answer: iii)Ho: μ₁=μ₂=μ₃, where 1,2 and 3 refer to country 1, country 2 and country 3.
Ha: not all population means are the same.

c) Give the value of the test statistic to three decimal places:

F= 266.058

d) Based on the ANOVA test, at the 1% significance level, we can (reject / fail to reject / accept) the null hypothesis.

Reject the null hypothesis.

e) Based on the ANOVA test, at the 1% significance level, we can conclude:

A)That the average heights of adult males in the three countries are all the same.

Answer: B)That the average heights of adult males in the three countries are not all the same.

C)That the average heights of adult males in the three countries are all different.

f) At the 1% significance level:

A)The average height of adult males in country 1 is significantly different from the average height of adult males in country 2.

Answer: B)The average height of adult males in country 1 is significantly different from the average height of adult males in country 3.

C)The average height of adult males in country 1 is significantly different from the average height of adult males in country 2 and in country 3.