Exam 11: Statistical Inferences for Population Variances

In testing the difference between two population variances,it is a common practice to compute the F statistic so that its value is always greater than or equal to one.

The value of χ²_α in a particular situation depends on:

In a sample of n = 16 selected from a normally distributed population,we find a population standard deviation of s = 10.What are the degrees of freedom for the hypothesis test?

The exact shape of the chi-square distribution depends on the degrees of freedom.

To compute a 95 percent confidence interval for σ²,we use n - 1 degrees of freedom and the chi-square points on the distribution curve of χ_α/2² and of χ_1-(α/2)².

In a sample of n = 16 selected from a normally distributed population,we find a population standard deviation of s = 10.What is the value of χ² if we are testing H₀: σ² = 144?

In order to make statistical inferences about σ² that are valid using a chi-square distribution,the assumption is that the sampled population is also a chi-square distribution.

When computing a 95 percent confidence interval for σ² with a sample of n = 30,we would use the following values of χ² in the calculations:

A baker must monitor the temperature at which cookies are baked.Too much variation will cause inconsistency in the texture of the cookies.Past records show that the variance of the temperatures has been 1.44°.A random sample of 30 batches of cookies is selected,and the sample variance of the temperature is 4.41°.What is the 95 percent confidence interval for σ² at α = .05?

In a sample of n = 25 selected from a normally distributed population,we find a population variance of s² = 150.What is the value of χ² if we are testing H₀: σ² = 100?

The chi-square distribution is a continuous probability distribution that is skewed to the left.

We use the following data for a test of the equality of variances for two populations at α = .10.Sample 1 is randomly selected from population 1 and sample 2 is randomly selected from population 2.Can we reject H₀ at α = .10?

When evaluating a new process,using the square root of the upper end of the confidence interval for σ² gives an estimate of the smallest that σ for the new process might reasonably be.

χ²_α is the point on the vertical axis under the curve of the chi-square distribution that gives a righthand tail area equal to α.

A manufacturer of an automobile part has a process that is designed to produce the part with a target of 2.5 inches in length.In the past,the standard deviation of the length has been 0.035 inches.In an effort to reduce the variation in the process,the manufacturer has redesigned the process.A sample of 25 parts produced under the new process shows a sample standard deviation of 0.025 inches.Calculate the test statistic for testing whether the new process standard deviation has improved from the current process.

Consider an engine parts supplier,and suppose the supplier has determined that the mean and variance of the population of all cylindrical engine part outside diameters produced by the current machine are,respectively,2.5 inches and .00075.To reduce this variance,a new machine is designed.A random sample of 20 outside diameters produced by this new machine has a sample mean of 2.5 inches and a variance of s² = .0002 (normal distribution).In order for a cylindrical engine part to give an engine long life,the outside diameter must be between 2.43 and 2.57 inches.Assuming normality,determine whether 99.73 percent of the outside diameters produced by the current machine are within specification limits.

When comparing the variances of two normally distributed populations using independent random samples,if $s _ { 1 } ^ { 2 } = s _ { 2 } ^ { 2 }$ the calculated value of F will always be equal to one.

Using a χ² test statistic to test the null hypothesis that the variance of a new process is equal to the variance of the current process and rejecting at p-value less than α,we can conclude that the new process is more consistent than the current process.

Consider an engine parts supplier,and suppose the supplier has determined that the mean and variance of the population of all cylindrical engine part outside diameters produced by the current machine are,respectively,2.5 inches and .00075.To reduce this variance,a new machine is designed.A random sample of 20 outside diameters produced by this new machine has a sample mean of 2.5 inches and a variance of s² = .0002 (normal distribution).In order for a cylindrical engine part to give an engine long life,the outside diameter must be between 2.43 and 2.57 inches.If σ² denotes the variance of the population of all outside diameters that would be produced by the new machine,test H₀: σ² = .00075 versus H_a: σ² < .00075 by setting α = .05.

Consider an engine parts supplier,and suppose the supplier has determined that the mean and variance of the population of all cylindrical engine part outside diameters produced by the current machine are,respectively,2.5 inches and .00075.To reduce this variance,a new machine is designed.A random sample of 20 outside diameters produced by this new machine has a sample mean of 2.5 inches,a variance of s² = .0002 (normal distribution).In order for a cylindrical engine part to give an engine long life,the outside diameter must be between 2.43 and 2.57 inches.Find the 95 percent confidence intervals for σ² and σ.