Question

 

Regression

Is there a relationship between the number of stories a building has and its height? Some statisticians compiled data on a set of n = 60 buildings reported in the World Almanac. You will use the data set to decide whether height (in feet) can be predicted from the number of stories.

data from buildings.txt.
(Note that this is a text file, so use the appropriate instruction. If you are having trouble uploading the data, open it to see its contents and type the data in: one vector for heights and one vector for stories. Ignore the year data.)

buildings.txt

YEAR   Height   Stories
1990   770   54
1980   677   47
1990   428   28
1989   410   38
1966   371   29
1976   504   38
1974   1136   80
1991   695   52
1982   551   45
1986   550   40
1931   568   49
1979   504   33
1988   560   50
1973   512   40
1981   448   31
1983   538   40
1968   410   27
1927   409   31
1969   504   35
1988   777   57
1987   496   31
1960   386   26
1984   530   39
1976   360   25
1920   355   23
1931   1250   102
1989   802   72
1907   741   57
1988   739   54
1990   650   56
1973   592   45
1983   577   42
1971   500   36
1969   469   30
1971   320   22
1988   441   31
1989   845   52
1973   435   29
1987   435   34
1931   375   20
1931   364   33
1924   340   18
1931   375   23
1991   450   30
1973   529   38
1976   412   31
1990   722   62
1983   574   48
1984   498   29
1986   493   40
1986   379   30
1992   579   42
1973   458   36
1988   454   33
1979   952   72
1972   784   57
1930   476   34
1978   453   46
1978   440   30
1977   428   21

1. Draw a scatterplot with stories in the x-axis and height in the y-axis. Describe the trend, strength and shape of the relationship between stories and height.

2. Find the linear correlation coefficient between these variables. How does it support the description you gave in (b)?

3. Draw diagnostic plots (a plot of stories vs. residuals, and a normal probability plot for the residuals). Do assumptions appear to be satisfied?

4. Obtain a 95% confidence interval for the true value of the slope. How does the interval support your conclusion in (e)?

5. What is the estimated height of a building that is 45 stories high? Write a concluding sentence supported by your results above.

~~~~~~~~~~~~(Please display all RCode)~~~~~~~~~~~~~~

Answer 1

> Height=c(770 , 677  ,428  ,410  ,371  ,504  ,1136  ,695  ,551  ,550  ,
+ 568  ,504  ,560  ,512  ,448  ,538  ,410  ,409  ,504  ,777  ,496  ,386  ,
+ 530  ,360  ,355  ,1250  ,802  ,741  ,739  ,650  ,592  ,577  ,500  ,469  ,
+ 320  ,441  ,845  ,435  ,435  ,375  ,364  ,340  ,375  ,450  ,529  ,412  ,
+ 722  ,574  ,498  ,493  ,379  ,579  ,458  ,454  ,952  ,784  ,476  ,453  ,
+ 440  ,428 )
>
> Stories=c(54,47,28,38,29,38,80,52,45,40,49,33,50,40,31,40,27,31,35,57,31,
+ 26,39,25,23,102,72,57,54,56,45,42,36,30,22,31,52,29,34,20,33,18,23,30,38,
+ 31,62,48,29,40,30,42,36,33,72,57,34,46,30,21)
>
>
>
> ## 1) Scatter plot
>
> plot(Stories~Height)
>
>
>
>
> #
> plot(Height~Stories)

Comment: The trend is used to predict future values based on recently observed data. But, this data does not appropriate to describe the trend. From the scatter plot, increase the value of stories increases the values of Height and vice versa. Hence, the strength is strong and the shape of the relationship is linear between stories and height.

#  the linear correlation coefficient between these variables is

> cor(Height, Stories)
[1] 0.9505549

Comment: The estimated correlation value is 0.9505549. It is a positive value and more than 0.5. Hence, the two variables have a strong positive association.

> ### Linera model
>
> model=lm(Height~Stories)
> res=residuals(model)
>
> plot(res, Stories)

Comment: From the above plot between the residuals and stories, we observed that if we remove around four points which is available at 0 to 90 degree, the relationship is slightly negatively linear. Whereas, as all the relationship is random and satisfied the independence between the residual and stories.

## Normal probability plot
qqnorm(res)
qqline(res)

From the normal probability plot for the residuals, the values of the residuals at both the tails are dispersed from the straight line. Hence, the residuals have a high peak, so the normal assumption on the residual may not be satisfied.

# The 95% confidence interval for the true value of the slope is

> confint(model)
2.5 % 97.5 %
(Intercept) 48.34928 132.26993
Stories 10.32267 12.26208

# The 95% confidence interval for the true value of the slope is (10.32267, 12.26208).

The 95% confidence interval does not include the value zero. Hence, we can conclude that stories have a significant effect on height at 0.05 level of significance.

## Prediction

> new.Stories<- data.frame(
+ Stories = c(45)
+ )
>
> predict(model, newdata =new.Stories, interval = "confidence")
fit lwr upr
1 598.4665 582.7431 614.1899

The estimated height of a building that is 45 stories high is 598.4665.

When increasing the stories high by 45 the mean estimated height is 598.36.