- Provide synonyms for the word
*variance*. What symbol is used to denote the population variance? What symbol is used to denote the sample variance? - Provide a synonym for the term
*standard deviation*. What symbol is used to denote the population standard deviation? What symbol is used to denote the sample standard deviation? - When will 95% of values lie within 2 standard deviations of the mean?
- What is the name of the rule that states "at least 75% of values lie within 2 standard deviations of the mean."
- Name two measures of spread other than the variance and standard deviation.
- How do you use a boxplot to assess variance?
- The sum of squares is the sum of the squared distances of data points around the group's ___________.
- State the relation between SS and variance.
- What procedure tests for inequality of population variances?
- Why
*not*pool variances when the*F*ratio test is significant? - T/F: The standard error of the mean difference is a measure of spread.
- When
*pooling*variances (for Student's*t*procedures),*n*_{1}= 11 and*n*_{2}= 10. Then, df_{1}= ____, df_{2}= ____, and df = ____. - Suppose
*t*_{stat }= 2.96 with 16 degrees of freedom. Is the two-sided*p*value for this problem less than 0.05? Is it less than 0.01? - What is the value of the
*t*quantile used to calculate a 95% confidence interval formula for µ_{1}- µ_{2}when*n*_{1}= 10 and*n*_{2}= 8? (Equal variance method.) - Using statistical notation, write the null and alternative hypotheses
for the
*F-*ratio test. - In words, write the null and alternative hypotheses for
the
*F*ratio test. - Using statistical notation, write the two-sided null
hypotheses for an independent
*t*test. - What symbol is used to denote the independent mean difference in the population? . . . in the sample?
- List ways to compare group variability.
- List ways to compare group averages.
- What is the name of the
*t*test that makes no assumptions about the equality of population variances?

**11.1 Comparing means depends on within group variability. **Whether an observed difference
in
means is surprising
depends on the variance within groups. This makes sense when one considers that
it
is more likely differences will
arise by chance when individuals within groups vary greatly. This is why we take
variance into account when comparing means. Consider the stemplots below. In both comparisons, group 1 has a mean of 70
and group 2 has a mean of 50. However, we are confident the difference observed
in Comparison B is real, while the observed difference in Comparison A might be
due to chance fluctuation. Conduct

Comparison A
0|8| 0|7|0 0|6|0 0|5|0 |4|0 |3|0 |2| |1| (x10) |
Comparison B
0|8| 000|7| 0|6|0 |5|000 |4|0 |3| |2| |1| (x10) |

**11.****2 ** * Leaves on a common
stem. *Plot the data sets listed below as side-by-side stemplots.
(Separate side-by-side boxplots for each comparison).
Based on these plots, compare group means and variances. (Calculations

`Comparison A: Group 1: 90, 70, 50, 30,
10 Group 2: 70, 60, 50, 40, 30
Comparison B: Group
1: 90, 80, 70, 60, 50 Group 2: 70, 60, 50, 40, 30
Comparison C: Group 1: 90, 70, 50, 30, 10 Group 2: 90, 80, 70, 60,
50`

**11****.****3 *** Linoleic acid and LDL cholesterol
*
A study tested the cholesterol-lowering potential
of dietary linoleic acid in mildly hypercholesterolemia subjects (Rassias et al.,
1990). Plasma cholesterol (mmol/m

6.0 | 6.4 | 7.0 | 5.8 | 6.0 | 5.8 | 5.9 | 6.7 | 6.1 | 6.5 | 6.3 | 5.8 |

A
different (fcctitious) group of
had the following values:

6.4 | 5.4 | 5.6 | 5.0 | 4.0 | 4.5 | 6.0 |

Data are stored online in ` rassias.sav`.

(A)Compare the groups with stemplots on a common stem. Discuss your findings.

(B)The mean of group 1 = 6.192 mmol/m^{3}. Its standard deviation = 0.392 mmol/m^{3}. Calculate by hand the mean and standard deviation of group 2.

(C)Test the variances for inequality with anFratio test.

(D)Test the means with attest.

(E)Summarize your analysis.

**11.****4
***
Particulate matter in air samples. * In a study of
air pollution, investigators
measured suspended particulate matter (µgms/m

Site 1: 68 22 36 32 42 24 28 38

Site 2: 36 38 39 40 36 34 33 32

(A)Create stemplots on a common stem to compare these two distributions. Use split stem-values for your plot. Discuss your findings.

(B)Calculate the means and standard deviations of the data from the two sites. How do these summary statistics complement your stemplot analysis?

(C)Test the variances for inequality with anFratio test. Include a statement of the null hypothesis.

(D)Would you use a pooled (equal variance)tprocedure to test of means? Explain your response.

(E)Test the means for inequality. Show all hypothesis testing steps.

(F)What is the most important finding in this analysis? Was the test of means revealing or obscuring?

**11.****5 *** Body weight and pituitary adenoma*. The standard deviation of
body weights in

**11.****6 *** Anxiety during hemodialysis. *Severe anxiety often accompanies chronic hemodialysis. To help counteract
this anxiety, a set of progressive relaxation
exercises was shown on videotape to a group of 38 hemodialysis patients. A control group of 23 patients viewed a set of neutral videotapes. Following
these interventions, the State-Trait Anxiety Inventory questionnaire was
administered to both groups. The treatment group had a the mean anxiety score of
33.42
(standard deviation
= 10.18). The control group had a mean score of 39.71
(standard deviation = 9.16) (Alarcon,
1982).

(A) Test the variances for a significant difference. Show all steps of the procedure. Remember to interpret your results.

(B) Now test the difference in the means for significance. Again, show all steps in the procedure.

(C) For question B, did you use an equal variance or unequal variancetprocedure? Justify use of the procedure that you did use.

**11.****7 **

Total heart weight (grams) | |||||||||||

Group 1 (heart failure) | 450 | 760 | 325 | 495 | 285 | 450 | 460 | 375 | 310 | 615 | 425 |

Group 2 (controls) | 245 | 350 | 340 | 300 | 310 | 270 | 300 | 360 | 405 | 290 |

**11.****8 *** Body weights of cadavers
with and without heart failure
* (Rosner, 1990, p.
35)

(A) Using SPSS, explore the groups with side-by-side plots (of your choice)

(B) Test variances for equality using either anFratio test or Levene's test. Show all hypothesis testing steps.

(C) TestH_{0}: µ_{1}= µ_{2}. Would you use an unequal variance or equal variancetprocedure in this instance?

Body weight (kgs) | |||||||||||

Group 1 (heart failure) | 54.6 | 73.5 | 50.3 | 44.6 | 58.1 | 61.3 | 75.3 | 41.1 | 51.5 | 41.7 | 59.7 |

Group 2 (controls) | 40.8 | 67.4 | 53.3 | 62.2 | 65.5 | 47.5 | 51.2 | 74.9 | 59 | 40.5 |

**11.9 Efficacy of echinacea in treating upper
respiratory infections (severity of symptoms)**. A randomized, double-blind, placebo-controlled
trial evaluated the herbal remedy

**11.10** ** Efficacy of echinacea in treating upper
respiratory infections (duration of symptoms)**. The echinacea study
introduced in the prior exercise also measured the duration of peak
symptoms in study subjects. The treatment group (

**11.11 The effect of calcium
supplementation on blood pressure **(Lyle
et al., 1987).

SUBJECT |
GROUP |
BEFORE |
AFTER |
DELTA |

1 |
1 |
107 |
100 |
7 |

2 |
1 |
110 |
114 |
−4 |

3 |
1 |
123 |
105 |
18 |

4 |
1 |
129 |
112 |
17 |

5 |
1 |
112 |
115 |
−3 |

6 |
1 |
111 |
116 |
−5 |

7 |
1 |
107 |
106 |
1 |

8 |
1 |
112 |
102 |
10 |

9 |
1 |
136 |
125 |
11 |

10 |
1 |
102 |
104 |
−2 |

11 |
2 |
123 |
124 |
−1 |

12 |
2 |
109 |
97 |
12 |

13 |
2 |
112 |
113 |
−1 |

14 |
2 |
102 |
105 |
−3 |

15 |
2 |
98 |
95 |
3 |

16 |
2 |
114 |
119 |
−5 |

17 |
2 |
119 |
114 |
5 |

18 |
2 |
114 |
112 |
2 |

19 |
2 |
110 |
121 |
−11 |

20 |
2 |
117 |
118 |
−1 |

21 |
2 |
130 |
133 |
−3 |

**11.12 ** * The effect of calcium
supplementation on blood pressure. *For the data in Exercise
11.11, test the decreases in blood pressure for a significant difference.

Key to Odd Numbered Problems**
**Key to Even Numbered Problems (may not be posted)