STATISTICS

Statistics for Civil and Environmental Engineering

Statistics for Civil and Environmental Engineering

Question 1: alloy.dat: (a) Produce a dot plot, a boxplot and a histogram of the data. Which of these displays do you think best illustrates the distribution of these observations?

From above scatter plot 0f alloy data set we observed that data points are scattered through out the region.

From the above Box plot of alloy data set we can observed that the shape of the data is positively skewed as the middle line is slightly above the mean which 850.

Histogram is the best illustrates the distribution of these observations as we can observed that it is more symmetric in shape.

(b)A normal distribution is proposed as a model for the hardness of these specimens. Do the observations support this model?

The normal probability plot (Chambers 1983) is a graphical technique for assessing whether or not a data set is approximately normally distributed.

The data are plotted against a theoretical normal distribution in such a way that the points should form an approximate straight line. Departures from this straight line indicate departures from normality.

The normal probability plot is a special case of the probability plot. We cover the normal probability plot separately due to its importance in many applications.

As the normal probability plot follows the straight line so we can say that our alloy data set follows normal distribution.

(c) Using the data to estimate the parameters of the normal distribution, calculate the probability that a specimen of this alloy will have hardness (i) less than 845 megapascals (ii) greater than 870 megapascals (iii) between 840 and 875 megapascals.

(i) less than 845 megapascals

Normal with mean = 850.73 and standard deviation = 12.75

x P( X <= x )

845 0.326567

(ii) greater than 870 megapascals

Normal with mean = 850.73 and standard deviation = 12.75

x P( X >= x )

870 0.934653

(iii) between 840 and 875 megapascals

Normal with mean = 850.73 and standard deviation = 12.75

x P( 840< x<875)

0.868476

(d)Calculate a 95% confidence interval for the mean hardness of the alloy, and interpret this interval. Hence test the hypothesis that the mean hardness ( is equal to 850 megapascals. Calculate a p-value for this test.

Hypothesis:

( = 850

( ? 850

One-Sample Z: Alloy

Test of mu = 850 vs not = 850

The assumed standard deviation = 12.75

Variable N Mean StDev SE Mean 95% CI Z P

Alloy 15 850.73 12.76 3.29 (844.28, 857.19) 1.74 0.082

As we observed that p value is .082 which is greater than .05 so we reject null hypothesis and conclude that ( ? 850

Question 2: The data in the file weld.dat were recorded by the Welding Institute in Abingdon. The data represent 21 measurements of the current used during welding (c1, in kiloamps) and the resulting diameter of the welded zone (c2, in mm).

(a)Plot the diameter against the current.

Scatter plots are similar to line graphs in that they use horizontal and vertical axes to plot data points. However, they have a very specific purpose. Scatter plots show how much one variable is affected by another. The relationship between two variables is called their ...