Question

1. Let X equal the weight (in grams) of a nail of the type that is used for making decks. Assume the distribution of X is N(8.78, 0.4²). Let ? be the mean of a random sample of weights of n = 16 nails.
   1. Sketch, on the same set of axes, the graphs of the probability density functions of X and of ?. Comment on these two distributions.
   2. Find P(8.58 < X < 8.98) and P(8.58 < ? < 8.98). Compare the results.
   3. Let s² be the sample variance of the 16 weights. Find the constants a and b such that P(a < s² < b) = 0.90.

Answer 1

a.)

The black line indicates the pdf of X

The red line indicates the pdf of Sample mean

b.)

P(8.58<X<8.98)=P(X<8.98)-P(X<8.58)

=.6211-..3788

=.2423

P(8.58<Xbar<8.98)=P(Xbar<8.98)-P(Xbar<8.58)

=.8914-.108522

=.78291

probability of sample mean (Xbar) is greater than probability of X

c.)

therefore

CV =CRITICAL VALUE

a=(CV at 5% *)/n-1)=..2033

b=(CV at 95% *)/n-1)=.6988