Exam 10: Hypothesis Testing: Additional Topics

The p-value of testing H₀ : P_x - P_y = 0 against H₁ : P_x - P_y ≠ 0 is the probability that the null hypothesis is true.

Assume that we have independent random samples of size n_x and n_yobservations from normal populations with means μ_x and μ_y and unequal variances.The sample variances and Are used and the means are And )Which of the following formulae calculates the degree of freedom,v,for the Student's t statistic where variance is unknown and considered unequal?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Are medical students more motivated than law students? A randomly selected group of each were administered a survey of attitudes toward Life,which measures motivation for upward mobility.The scores are summarized below. -Judging from the way the data were collected,which test would likely be most appropriate to employ?

The statistical distribution that is used for testing the difference between two population variances is the:

Suppose that a t test of H₀ : μ₁ - μ₂ = 0 against H₁ : μ₁ - μ₂ ≠ 0 is based on 15 degrees of freedom.If the calculated value of the test statistic is t = -1.75,then the p-value for this test is approximately 0.10.

When testing for differences between the means of two related populations,either a one-tailed or two-tailed test can be used.

In testing for differences between the means of two independent populations if the test statistic is very large,then the p-value of the test should be very small.

The F distribution is constructed as the ratio of two Student's t random variables;each divided by its degrees of freedom.

A null hypothesis of the difference between two population proportions is rejected at the 5% level,but not at the 2.5% level.This means that the p-value of the test is between 0.025 and 0.05.

In practical applications,the F ratio is arranged in a way such that the larger sample variance is in the denominator and the smaller is in the numerator.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: A trap and skeet shooting club,purchases clay pigeons.These are quite fragile and often break in the box during shipment without any outward sign of damage.Even as the clay pigeons are loaded into the throwing machine,broken or cracked clay pigeons may not be noticed.Each clay pigeon broken is a net loss to the club so the club is anxious to purchase from suppliers who have a record of little or no breakage in shipping and handling.Two suppliers of clay pigeons are contacted and asked for price quotes.Supplier 1 quotes a higher price than Supplier 2 but says that the higher price is the result of better,more expensive packaging and thus results in more usable clay pigeons per case.The shooting club manager decides to purchase 10 cases from each supplier.Each case has 250 clay pigeons.Careful records are maintained on each case as to the number of good clay pigeons.The records after the cases were used are as follows: Number of good clay pigeons -Use Excel or Minitab to summarize this data using descriptive statistics.

When the sample sizes are equal,the pooled variance of the two groups is the average of the two sample variances.

What conclusion can be made about the effectiveness of the diet,based on the test,at the 0.05 level of significance?

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: As part of its quality control program,the Scullia Auto Company (SAC)began inspecting all supplier shipments of steel bars and recording the defect rate according to quality criteria such as strength,dimensional characteristics,etc.When SAC began this program,the defect rate in 100 shipments averaged 9%.A recent report indicated an average defect rate of 2.2% in 136 shipments.The quality control department wants to determine,at the 0.05 level of significance,if this represents a significant improvement in the quality of the steel. -Calculate an estimate of the common proportion.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: A manager is considering the purchase of new production equipment to improve the output of the plant.Suppose the equipment from two suppliers is extensively tested in multiple production runs.A total of 20 runs are recorded on Supplier A's equipment and another 15 runs are reported from Supplier B's equipment.The volume produced is recorded for each of the 35 runs.The results are summarized in the table below. -The numerator and denominator degrees of freedom associated with the test statistic for determining if there is a difference in the population variances for Supplier A and B are,respectively:

At 5% significance level,the upper-tailed critical value of an F test with 3 degrees of freedom in the numerator and 8 degrees of freedom in the denominator is equal to 4.07.

Based on the given information,the p-value for the F test of equal variances can be calculated and shown to be 0.289.Based on this p-value,what is your conclusion?

From a random sample of 86 health insurance firms,61 did public relations in-house,as did 55 of an independent random sample of 86 casualty insurance firms.Test at the 5% significance level the equality of the population proportions against a two-sided alternative.

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: The following data were generated from a matched pairs experiment. -Use statistical software to determine whether these data are sufficient to infer at the 10% significance level that the two population means differ.

Suppose that the test statistic is Z = 1.95.Find the p-value if we assume that the alternative hypothesis is a two-tailed test (H₁ : μ_M - μ_L ≠ 0).