Question

The United States Army recently commissioned a study to assess how deeply a bullet penetrates ceramic body armor. In the study, a cylindrical clay model was layered under an armor vest. A projective was then fired, causing an indentation in the clay. The deepest impression in the clay was measured as an indication of survivability of someone wearing the armor. Here are the data that were observed; measurements (Y , measured in mm) were made using a manually controlled digital caliper. The data are already ordered from low to high.

y <- c(22.4,23.6,24.0,24.9,25.5,25.6,25.8,26.1,26.4,26.7,27.4,27.6,28.3, 29.0,29.1,29.6,29.7,29.8,29.9,30.0,30.4,30.5,30.7,30.7,31.0,31.0, 31.4,31.6,31.7,31.9,31.9,32.0,32.1,32.4,32.5,32.5,32.6,32.9,33.1, 33.3,33.5,33.5,33.5,33.5,33.6,33.6,33.8,33.9,34.1,34.2,34.6,34.6, 35.0,35.2,35.2,35.4,35.4,35.4,35.5,35.7,35.8,36.0,36.0,36.0,36.1, 36.1,36.2,36.4,36.6,37.0,37.4,37.5,37.5,38.0,38.7,38.8,39.8,41.0, 42.0,42.1,44.6,48.3,55.0)

(a) Check the normal assumptions with plots. State clearly why do you use the plots and what does the plot imply.

(b) Construct a 95 percent confidence interval for the population mean µ and interpret your interval. Explain clearly (in words) what µ represents in the context of this problem.

(c) A co-worker of yours looks at your interval in part (a) and says, ”I’m confused. Ninetyfive percent of the measurements above should be in this interval, but far fewer are.” How would you respond to him?

(d) Calculate a 95 percent confidence interval for the population standard deviation σ and interpret your interval. Explain clearly (in words) what σ represents in the context of this problem.

Answer 1

(a) The plot is necessary to check if the distribution is normal. We would analyze this as below:

Since a clear increasing pattern is followed without any outlier, we can say that the data is normally distributed.

(b) µ represents measurements of how deeply a bullet penetrates ceramic body armor.

The 95 percent confidence interval for the population means µ is between 32.21951 and 34.52025.

(c) We would say we are 95% confident that the true mean measurement is between 32.21951 and 34.52025.

(d) σ represents the population standard deviation for measurements of how deeply a bullet penetrates ceramic body armor.

The 95 percent confidence interval for the population standard deviation is between 4.57 and 6.22.

The R code is:

y <- c(22.4,23.6,24.0,24.9,25.5,25.6,25.8,26.1,26.4,26.7,27.4,27.6,28.3, 29.0,29.1,29.6,29.7,29.8,29.9,30.0,30.4,30.5,30.7,30.7,31.0,31.0, 31.4,31.6,31.7,31.9,31.9,32.0,32.1,32.4,32.5,32.5,32.6,32.9,33.1, 33.3,33.5,33.5,33.5,33.5,33.6,33.6,33.8,33.9,34.1,34.2,34.6,34.6, 35.0,35.2,35.2,35.4,35.4,35.4,35.5,35.7,35.8,36.0,36.0,36.0,36.1, 36.1,36.2,36.4,36.6,37.0,37.4,37.5,37.5,38.0,38.7,38.8,39.8,41.0, 42.0,42.1,44.6,48.3,55.0)
qqnorm(y)
t.test(y, level = 0.95)