Appendix F — Practice Problem Solutions — Chapter 8

The solutions use the following imports:

import numpy as np
import pandas as pd
import scipy.cluster as cluster
import scipy.stats as stats
import seaborn as sns
import statsmodels.formula.api

Solution Section 8.7.1

np.random.seed(2503478)
group_A = stats.norm(loc=10, scale=2).rvs(30)
group_B = stats.norm(loc=15, scale=2).rvs(30)
result = stats.ttest_ind(group_A, group_B)
print(result)

TtestResult(statistic=np.float64(-8.473531241332175), pvalue=np.float64(9.899983754486164e-12), df=np.float64(58.0))

Solution Section 8.7.2

np.random.seed(493567)
group_A = stats.norm(loc=10, scale=2).rvs(30)
group_B = stats.norm(loc=15, scale=2).rvs(30)
group_C = stats.norm(loc=11, scale=2).rvs(30)
result = stats.f_oneway(group_A, group_B, group_C)
print(result)

F_onewayResult(statistic=np.float64(56.614548038664914), pvalue=np.float64(1.7891834179400852e-16))

Solution Section 8.7.3

result = stats.tukey_hsd(group_A, group_B, group_C)
print(result)

Tukey's HSD Pairwise Group Comparisons (95.0% Confidence Interval)
Comparison  Statistic  p-value  Lower CI  Upper CI
 (0 - 1)     -4.656     0.000    -5.787    -3.525
 (0 - 2)     -0.643     0.369    -1.774     0.488
 (1 - 0)      4.656     0.000     3.525     5.787
 (1 - 2)      4.013     0.000     2.882     5.144
 (2 - 0)      0.643     0.369    -0.488     1.774
 (2 - 1)     -4.013     0.000    -5.144    -2.882

Solution Section 8.7.4

np.random.seed(932847)
x1 = np.random.uniform(-10, 10, 50)
x2 = np.random.uniform(-10, 10, 50)
y = 3 * x1 + np.random.normal(0, 2, 50)
df = pd.DataFrame({"X1": x1, "X2": x2, "Y": y})
model = statsmodels.formula.api.ols("Y ~ X1 + X2", data=df).fit()
print(model.summary().tables[1])

==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     -0.4118      0.295     -1.395      0.170      -1.006       0.182
X1             3.0064      0.046     65.950      0.000       2.915       3.098
X2            -0.0319      0.053     -0.606      0.547      -0.138       0.074
==============================================================================

Solution Section 8.7.5

Two clusters mainly separate setosa from versicolor and virginica, which makes biological sense given that setosa is the most distinct species.

iris = pd.read_csv("../../_data/iris.csv")
sns.pairplot(iris, hue="Species")
_centroids, labels = cluster.vq.kmeans2(iris.drop(columns="Species"), k=2)
sns.pairplot(iris.assign(Cluster=labels), hue="Cluster")