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MA 2612 A' 98 Test 2

A science historian has collected data on the evolution of the efficiency of steam engines over the years 1700-1955, and found that the log base 10 of efficiency seems linearly related to year. Figure 1 is SAS/INSIGHT output for the regression of the log base 10 of efficiency on year for 11 different years during this period.

What is the fitted regression line? Interpret the fitted slope. Is it reasonable to interpret the intercept? If so, interpret it. If not, tell why not.


\widehat{LOG10EFF}=-13.4515+0.0067 YEAR \mbox{\bf (5

The change in predicted LOG10EFF is 0.0067 per year. (5 points) The intercept is well beyond the range of the data, and as such it is not reasonable to interpret it. In fact, it is not clear what steam technology, as we know it, existed in the year 0. (5 points)

By what percentage does use of this model reduce the uncertainty in predicting the log of efficiency?

Answer: 99.26% (10 points)

What is the value of the Pearson correlation coefficient between the log of efficiency and year?

Answer: 0.9963 (10 points)

Estimate $\sigma$, the standard deviation of the random errors.

Answer: $\hat{\sigma}=0.0531$ (10 points)

The standard deviation of the 11 values of YEAR in the data set is 86.4875. Use this, the output in Figure 1, and your knowledge of regression to obtain the standard deviation of the 11 values of LOG10EFF in the data set.

Answer: We know that


Therefore, SY=0.5816. (5 points)

Figure 1:   Output from SAS/INSIGHT regression of log10 efficiency on year, problem 1.

A random sample of 81 children was obtained to study whether handedness is gender-related. The data are shown in Table 1.

Table 1:   Gender versus handedness for 81 children
  Boys Girls Total
Left-handed 9 11 20
Right-handed 29 32 61
Total 38 43 81

Summarize these data using overall, row and column percentages.

Row Pct      
Col Pct      
  Boys Girls Total
Left-handed 9 11 20
  11.11 13.58 24.69
  45.00 55.00  
  23.68 25.58  
Right-handed 29 32 61
  35.80 39.51 75.31
  47.54 52.46  
  76.32 74.42  
Total 38 43 81
  46.91 53.09 100.00

What are the marginal distributions of gender and handedness?

Gender: 46.91% boys, 53.09% girls. Handedness: 24.69% left, 75.31% right.

What is the conditional distribution of handedness given the individual is male? Female?

Females: 25.58% left, 74.42% right. Males: 23.68% left, 76.32% right.

Compute the Pearson residual in each cell under the assumption that gender and handedness are independent. Do any cells have perticularly large Pearson residual values?

For the 1,1 cell, the expected frequency is $(20\times
38)/81=9.38$, and the Pearson residual is $(9-9.38)/\sqrt{9.38}=-0.12$. The Pearson residual for the 1,2 cell is 0.12, for the 2,1 cell is 0.07, and for the 2,2 cell is -0.07. None of these values is large.

Conduct a $\chi^2$ test for the independence of gender and handedness at the 0.05 level of significance. What do you conclude?

The value of the $\chi^2$ test statistic is -0.122+0.072+0.122+(-0.072)=0.0386, which has a p-value of 0.843 (Or, note that 0.0386 is less than $\chi^2_{1,0.95}=3.841$). So there is little evidence of a relation between gender and handedness.

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Joseph D Petruccelli