We take a random sample of bone density measurements from 16 American college fe

Question

We take a random sample of bone density measurements from 16 American college females. The measurements are as follows (In gm/cm3): 1.085, 1.041, 1.075, 1.034, 1.050, 1.055, 1.070, 1.090 1.072, 1.083, 1.081, 1.051, 1.051, 1.048, 1.051, 1.033 It is reasonable to treat these measurements as coming from a normal distribution with unknown mean mu and unknown variance sigma^2 a)Use the data to calculate an unbiased point estimate of the true mean bone density, mu, of American College females. b) Use the data to find an unbiased point estimate of the population variance, sigma^2 of bone density of American College females. c) Find the sample standard deviation of the above data d) Create a 90% confidence interval for mu. e) What critical value did you use to calculate the 90% confidence interval in d? f) Create a 90% prediction interval for mu. g) Copy your R script for the above into the text box here.

Explanation / Answer

The complete R snippet is as follows

df<- c(1.085,1.041,1.075,1.034,1.050,1.055,1.070,1.090,1.072,1.083,1.081,1.051,1.051,1.048,1.051,1.033)

## mean

mean(df)

## variance

sd(df)^2

## standard deviation of the sample

sd(df)/sqrt(length(df))

## 90% confidence interval

mean(df) + 1.644*sd(df)/sqrt(length(df)) ## upper limit

mean(df) - 1.644*sd(df)/sqrt(length(df)) ### lower limit

## the critical value is 1.645
##using

qnorm(1-(1-0.90)/2)

############

The results are

> df<- c(1.085,1.041,1.075,1.034,1.050,1.055,1.070,1.090,1.072,1.083,1.081,1.051,1.051,1.048,1.051,1.033)
> ## mean
> mean(df)
[1] 1.060625
> ## variance
> sd(df)^2
[1] 0.00034905
> ## standard deviation of the sample
> sd(df)/sqrt(length(df))
[1] 0.00467072
> ## 90% confidence interval
> mean(df) + 1.644*sd(df)/sqrt(length(df)) ## upper limit
[1] 1.068304
> mean(df) - 1.644*sd(df)/sqrt(length(df)) ### lower limit
[1] 1.052946
> ## the critical value is 1.645
> ##using
> qnorm(1-(1-0.90)/2)
[1] 1.644854

Please note that we can answer only 4 subparts of a question at a time , as per the answering guidelines. I have answered 5 here

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