The following problem is partially completed, need explanation of a) and b) below:

"An engineering team believed that the underlying distribution for a process was a Beta distribution with shape parameters 1 and 0.5. They wanted to conduct a quick goodness of fit check using 30 observations to see if the distribution they chose was reasonable. Prior to running the test, one engineer questioned if the Chi-Square Goodness of fit test using intervals of [0, .2), [.2, 4), [.4, .6), [.6, .8), [.8, 1] would be sensitive enough to detect a difference even if the actual distribution appeared considerably different, for example, a Beta distribution with shape parameters 1 and 0.95. Assume that detecting a difference is a test significant at = 0.05."

Compare the two distributions, just turn in the single plot generated by:

> curve(dbeta(x,1,.5),from=0,to=1,col="black")

> curve(dbeta(x,1,.95),from=0,to=1,col="red",add=T)

Generate 40 observations from the wrong distribution:

> set.seed(13455)

> data<-rbeta(40,1,.95)

Get the cumulative probabilities for the hypothesized distribution

> pbeta(c(0.2, 0.4, 0.6, 0.8, 1), 1, 0.5)

Result:

[1] 0.1055728 0.2254033 0.3675445 0.5527864 1.0000000

Now:

a) Find the interval probabilities.

b) Conduct the appropriate Chi-Square test using the 5 equal intervals over X and comment on the sensitivity of the test for = 0.05.