1. The following statistics are taken from an article by P. Burch relating cigarette smoking to lung cancer. The article presents data relating mortality from lung cancer to average cigarette consumption (lb/person) for females in England and Wales over a 40-year period.

The data are given in the table below.

Cigarette consumption and lung-cancer mortality in England and Wales, 1930-1969

1. Compute the correlation between 5-year lung cancer mortality and annual cigarette consumption when each is expressed in the \[{{\log }_{10}}\] scale.
2. Test this correlation for statistical significance, and report a p-value.
3. Fit a regression line relating 5-year lung-cancer mortality to annual cigarette consumption.
4. To test the significance of this regression line, is it necessary to perform any additional tests other than those in 1.2? If so perform them.
5. What is the expected lung-cancer mortality rate with an annual cigarette consumption of 1 lb/person?
6. Why are the variables mortality rate and annual cigarette consumption expressed in a log scale?

2. A 65 year old woman with a low bone density in 1992 was treated with alendronate through the year 1999. Bone density was measured irregularly over this period. The results for change in bone density of the lumbar spine are shown in table 1.

Change in bone density, lumbar spine, over time

Note: \[\sum\limits_{i=1}^{8}{{{t}_{i}}}\] =375, \[\sum\limits_{i=1}^{8}{{{x}_{i}}=6.69}\] , \[\sum\limits_{i=1}^{8}{t_{i}^{2}}\] =25,849, \[\sum\limits_{i=1}^{8}{x_{i}^{2}}\] =5.60197, \[\sum\limits_{i=1}^{8}{{{x}_{i}}{{t}_{i}}}\] =320.874

2.1 What is the estimated rate of increase in bone density of the lumbar spine per year? What is the standard error of the estimated rate of increase per year?

2.2 Provide a significance test to assess whether the mean bone density has significantly changed over time. Please note a 2-tailed p-value.

2.3 The normal change in bone density over time from age 40 to age 80 is a decrease of 0.15 gm/ \[c{{m}^{2}}\] . Does the rate of change in this woman differ significantly from the expected age-related change?

Another parameter measured was bone density at the femoral neck (hip). The results are shown in table 2.

Change in bone density, femoral neck over time

Note: \[\sum\limits_{i=1}^{8}{{{y}_{i}}}\] =5.250, \[\sum\limits_{i=1}^{8}{y_{i}^{2}}\] =3.4475, \[\sum\limits_{i=1}^{8}{{{x}_{i}}{{y}_{i}}}\] =4.3922

2.4 Provide a measure of association between bone density of the lumbar spine and bone density of the femoral neck.

2.5 Assess whether there is a significant relationship between bone density of the lumbar spine and bone density of the femoral neck. Please provide a two-tailed p-value.

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