diff --git a/core/modules/simpletest/tests/upgrade/upgrade.test b/core/modules/simpletest/tests/upgrade/upgrade.test
index ffb3a55..0ae6462 100644
--- a/core/modules/simpletest/tests/upgrade/upgrade.test
+++ b/core/modules/simpletest/tests/upgrade/upgrade.test
@@ -312,7 +312,9 @@ abstract class UpgradePathTestCase extends DrupalWebTestCase {
     $this->refreshVariables();
     $this->checkPermissions(array(), TRUE);
 
-    return TRUE;
+    // Check we can load the front page of the new site.
+    $this->drupalGet('');
+    return $this->assertText('Powered by Drupal', t('The front page of the upgraded site loads successfully.'));
   }
 
   /**
diff --git a/core/modules/simpletest/tests/upgrade/upgrade_bare.test b/core/modules/simpletest/tests/upgrade/upgrade_bare.test
index db71228..f507aea 100644
--- a/core/modules/simpletest/tests/upgrade/upgrade_bare.test
+++ b/core/modules/simpletest/tests/upgrade/upgrade_bare.test
@@ -27,5 +27,8 @@ class BareUpgradePathTestCase extends UpgradePathTestCase {
    */
   public function testBareUpgrade() {
     $this->assertTrue($this->performUpgrade(), t('The upgrade was completed successfully.'));
+
+    // Ensure that the new Entity module is enabled after upgrade.
+    $this->assertTrue(module_exists('entity'), 'Entity module enabled after upgrade.');
   }
 }
diff --git a/core/modules/statistics/statistics.test b/core/modules/statistics/statistics.test
index a014767..f12490a 100644
--- a/core/modules/statistics/statistics.test
+++ b/core/modules/statistics/statistics.test
@@ -130,7 +130,7 @@ class StatisticsLoggingTestCase extends DrupalWebTestCase {
     $this->assertEqual(array_intersect_key($log[6], $expected), $expected);
 
     // Create a path longer than 255 characters.
-    $long_path = $this->randomString(256);
+    $long_path = $this->randomName(256);
 
     // Test that the long path is properly truncated when logged.
     $this->drupalGet($long_path);