Question

You 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 1.519 is smaller than 2.257

Yes, because 1.735 is smaller than 2.885

Yes, because 2.226 is smaller than 2.885

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: ?

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 reject/fail to reject/accept 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 not significantly different from the average height of adult males in country 2.

The average height of adult males in country 1 is not 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
173.9065,174.4416,173.3905
175.8508,172.408,171.8004
169.7117,176.3606,170.7854
174.3933,173.768,172.0378
173.5781,175.2501,171.3748
171.1385,174.2832,169.5553
172.4496,175.5553,172.5238
173.6139,174.1995,171.5606
178.4676,175.8349,171.3028
177.2542,176.209,171.8729
171.0752,177.6678,171.4921
177.6524,174.8088,169.6878
174.1881,171.8925,170.8565
173.0054,173.8406,170.8809
174.1721,177.1319,170.302
172.7926,173.4918,173.6232
172.9138,176.5414,169.7864
175.3345,175.2861,170.3743
175.2136,177.255,170.032
175.2258,172.1587,169.3387
174.1072,174.8035,170.8116
171.2888,173.2882,170.6889
174.1759,179.462,173.3951
175.5454,176.3378,170.7265
173.8333,177.1685,169.5037
174.652,173.5127,173.5052
174.1903,174.3971,172.952
172.6448,174.6913,170.7556
173.5408,176.7476,168.8408
171.9191,174.6832,170.4331
174.4326,176.1523,170.8661
171.3794,172.0223,171.5141
171.4967,174.5692,170.7083
171.8858,173.8646,171.7651
168.6836,172.7344,171.6878
175.2576,175.862,169.224
173.5878,175.523,169.6781
171.9676,175.1502,169.9883
175.1554,173.0995,170.3383
170.5327,176.7912,170.6191
172.9466,175.6253,171.1187
172.7378,174.6514,166.5562
173.5788,175.1343,170.4145
173.5693,174.707,172.9637
171.8027,172.4747,169.4999
173.0549,174.6715,172.5006
172.8527,173.853,171.6255
174.0416,173.6312,171.0565
174.7399,173.3654,171.3737
174.7639,174.2997,168.7524
171.8045,172.096,170.9732
173.216,176.5463,173.5059
171.2788,175.8801,171.2475
171.4297,175.07,171.1621
173.0897,175.0478,169.9987
175.3989,173.9028,171.0538
171.9455,176.628,171.4485
173.6571,174.9002,171.7396
172.7616,174.0282,170.5408
174.776,177.1271,170.7453
171.4664,174.7628,174.1355
173.1488,174.2165,167.7125
173.9288,174.6594,174.4442
174.7509,173.8281,171.6063
175.4163,173.4198,172.6001
173.2289,178.889,168.6038
170.8626,177.5832,170.2149
171.9865,175.5613,170.6829
171.5076,173.2143,171.7341
176.6257,173.8018,168.5947
172.1766,174.8352,171.8075
174.2221,176.2871,169.2807
172.8114,173.102,171.1993
174.4329,171.6052,172.0785
171.9527,172.9264,171.5906
170.9966,175.6003,172.724
170.9664,175.687,172.6091
173.8323,175.7775,170.1236
172.8339,174.9046,171.4856
172.8059,175.3755,169.6834
175.229,174.3858,169.1173
173.5374,176.393,172.4872
173.3967,173.0575,171.1001
175.4815,175.7825,171.0176
171.8933,173.8269,172.4667
174.1449,174.5977,171.9919
174.3526,175.9292,171.6253
172.7344,176.6586,172.9754
173.4235,173.4235,172.4947
171.3512,176.991,171.4596
171.3781,176.0902,170.0645
173.2573,174.9982,170.1227
174.1834,174.8072,172.8882
176.9782,174.7736,168.6823
172.0997,174.6453,171.0633
173.381,175.1346,168.1767
172.9763,175.1769,172.6307
170.2005,176.3391,172.3926

Answer 1

Let denote the mean heights of the adult males in the countries 1,2 and 3.

To test: Vs   Not all means are equal

Using SPSS, Running a One way ANOVA,

We get the output:

a) Based on the output of Levene's test for homogeneity of variance, we test:

Variances are homogeneous and hence, can be pooled. Vs H_a:  Variances are not homogeneous and hence, cannot be pooled

The levene's test statistic 1.521 is less than the critical value (at alpha = 0.01) 2.257.

a) The variances be pooled based on this data set because 1.519 is smaller than 2.257.

b) As mentioned above, the null and alternative hypothesis for the ANOVA test could include:

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) From the output, the value of the test statistic is F = 148.797

d) Based on the ANOVA test, at the 1% significance level, since the p-value of the test 0.001 < 0.01, the test is found to be significant. We may reject 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 not all the same.

To find out which pair of means differ, we may run a post hoc test:

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 and in country 3.

In fact, all three means differ from each other. No pair of population means are equal based on this data.