Question

Sandpaper is rated by the coarseness of the grit on the paper. Sandpaper that is more coarse will remove material faster. Jobs such as the final sanding of bare wood prior to painting or sanding in between coats of paint require sandpaper that is much finer. A manufacturer of sandpaper rated220, which is used for the final preparation of bare wood, wants to make sure that the variance of the diameter of the particles in their220sandpaper does not exceed2.0micrometers.41randomly selected particles are measured. The variance of the particle diameters is2.10micrometers. Assume that the distribution of particle diameter is approximately normal.

a. Construct the90%confidence intervals for the population variance and standard deviation.

Round your answers to two decimal places.

Confidence interval for the population variance: Enter your answer; confidence interval for the population variance, lower bound to Enter your answer; confidence interval for the population variance, upper bound.

Confidence interval for the population standard deviation: Enter your answer; confidence interval for the population standard deviation, lower boundto Enter your answer; confidence interval for the population standard deviation, upper bound.

b. Test at a5%significance level whether the variance of the particle diameters of all particles in220-rated sandpaper is greater than2.0micrometers.

Conclude that the population variance is Choose you answer in accordance to the item b) of the question statement                                                          not greater than2.0square micrometers.

Answer 1

a)

Sample Size,n=41

sample variance,s² = 2.10

Confidence Level,CL=0.9

Degrees of Freedom,DF=n-1 = 40

alpha,α=1-CL=0.1

alpha/2 ,α/2=0.05

Lower Chi-Square Value=χ²_1-α/2    =26.5093

Upper Chi-Square Value=χ²_α/2       =55.7585

confidence interval for variance is

lower bound=(n-1)s²/χ²^α/2 =1.5065

upper bound=(n-1)s²/χ²_1-α/2 =3.1687

confidence interval for std dev is  

lower bound=√(lower bound variance)=1.227395

upper bound=√(upper bound of variance=1.7801

b)

Ho :σ² =2

Ha :σ² >2

  

Level of Significance , α = 0.05

sample Std dev , s² = 2.1

Sample Size ,n = 41

  

Chi-Square StatisticX² = (n-1)s²/σ² = 42

  

degree of freedom,DF=n-1 = 40

  

one tail test  

p-Value=0.384

p-value >α=0.05,

Do not reject the null hypothesis

so, there is not enough evidence to conclude that variance of the particle diameters of all particles in220-rated sandpaper is greater than2.0micrometers at α=0.05