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## STATS 120C University of California Week 6 Probability Statistics Questions

### Question Description

1. Consider the one-way unbalanced groups ANOVA model

yij = μi ijiid 2

with i = 1,…,I; j = 1,…,Ji and εij N(0,σ ).
Assume the I groups are independent and ∑Ii=1 αi = 0. Consider the contrastL = ∑Ii=1 ciμi, and suppose we want to test H0 : L = 0 versus the alternative

H1 :LConsider the test statistics

where Lˆ = ∑Ii=1 ciȳi·, MSE denotes the Mean Square Error in the ANOVA table associated with the I groups of the one-way ANOVA model. Show that under the null hypothesis H0 : L = 0, the test statistic t follows a t– distribution with ∑Ii=1 Ji I degrees of freedom.

2. Arehabilitationcenterresearchwasinterestedinexaminingtherelationship between physical fitness prior to surgery of persons undergoing corrective knee surgery and time required in physical therapy until successful reha- bilitation. Patient records in the rehabilation center were examined, and 24 male subjects ranging in age from 18 to 30 years had undergone simi- lar corrective knee surgery during the past year were selected for the study. The number of days required for succesful completion of physical therapy and the prior physical fitness status (below average, average, above average, etc.) for each patient follows:

Observation ( j) Pre-surgerystatus 1 2 3 4 5 6 7 8 9 10

Lˆt=Ic2i

MSE·∑i=1 Ji

Poor BelowAverage Average AboveAverage Excellent

443843484447 29 42 38 40 43 40 303539283131 26 32 21 20 23 22 141822211720

`42  45  45  4330  4229  35  29  33`

20 28

(a) Suppose that we wish to determine whether or not the mean rehabil- itation time in patients with a ’Poor’ or ’Below Average’ pre-surgery health status is different from the mean rehabilitation time in patients in any of the other pre-surgery groups. State the appropriate null and alternative hypothesis for this question.

1. (b) Carry out a test of the hypothesis stated in (a). State what test statistic you are using, and provide all the necessary information used to cal- culate the test statistic (e.g. show ALL your work). Report the value of the test statistic, calculate the p-value, state your decision, and in- terpret the results of your test in a way that would be understandable to a non-statician.
2. (c) Form simultaneous 95% confidence intervals for the difference in the mean rehabilation time of:
1. patients with a ’Poor’ pre-surgery health status versus that a pa- tients with a ’Average’ pre-squrger health status;
2. patients with a ’Poor’ pre-surgery health status versus that a pa- tients with a ’Above Average’ pre-squrger health status;
3. patients with a ’Poor’ pre-surgery health status versus that a pa- tients with a ’Excellent’ pre-squrger health status;

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