Exam 8: Inference for Proportions

At a large Midwestern university,a simple random sample of 100 entering freshmen in 1993 found that 20 of the sampled freshmen finished in the bottom third of their high school class.Admission standards at the university were tightened in 1995.In 1997,a simple random sample of 100 entering freshmen found that only 10 finished in the bottom third of their high school class.Let p₁ and p₂ be the proportions of all entering freshmen in 1993 and 1997,respectively,who graduated in the bottom third of their high school class.Is there evidence that the proportion of freshmen who graduated in the bottom third of their high school classes in 1997 was reduced as a result of the tougher admission standards adopted in 1995,compared to the proportion in 1993? To determine this,test the hypotheses H₀: p₁ = p₂ versus H_a: p₁ > p₂.Compute the value of the z statistic for testing these hypotheses.What is the value of the P-value?

It is estimated that 780,000 surgical site infections (SSIs)occur each year.SSIs are the second most common type of healthcare-associated infections in U.S.hospitals and account for an extra $3.5 to $10 billion in healthcare costs per year.The national SSIs rate is 1.9%.A Georgetown medical office was interested in determining if their SSI rate were smaller than the national average.Out of a sample of 277 patients in their study,only one infection occurred.Is the Georgetown medical office SSIs rate less than the national average SSIs rate? (Note: Use a significance level of .05. )

A recent study used umbilical cord blood to test for 25 hydroxyvitamin D,which is an indicator of vitamin D status of the baby.It was reported that 65% of babies tested were deficient in vitamin D in spite of the fact that the mothers consumed vitamin D supplements during pregnancy.A researcher in a northern region felt that this percentage was too high for this region because with the reduced hours of sunshine during winter months,pregnant women tended to use higher doses of supplements to compensate.A sample of 125 newborns was tested,and 72 were declared to be deficient in vitamin D.What is the P-value and the conclusion using  = 0.05?

A recent study used umbilical cord blood to test for 25 hydroxyvitamin D,which is an indicator of vitamin D status of the baby.It was reported that 65% of babies tested were deficient in vitamin D in spite of the fact that the mothers consumed vitamin D supplements during pregnancy.A researcher in a northern region felt that this percentage was too high for this region because with the reduced hours of sunshine during winter months,pregnant women tended to use higher doses of supplements to compensate.A sample of 125 newborns was tested,and 72 were declared to be deficient in vitamin D.What would be the appropriate null and alternative hypotheses that the researcher should establish for a test of significance?

A simple random sample of 100 bags of tortilla chips produced by Company X is selected every hour for quality control.In the current sample,18 bags had more chips (measured in weight)than the labeled quantity.The quality control inspector wishes to use this information to calculate a 90% confidence interval for the true proportion of bags of tortilla chips that contain more than the label states.What is the value of the standard error of ?

The candy company that makes M&M's claims that 10% of the M&M's it produces are green.Suppose that the candies are packaged at random,and the small bags contain 25 M&M's.When we randomly pick a bag of M&M's,we may assume that this represents a simple random sample of size n = 25.Suppose that in a randomly selected small bag of M&M's,there are 5 green M&M's.At the 1% significance level,are the data statistically significant for testing if the true proportion of green M&M's is indeed equal to 10%?

A noted psychic was tested for extrasensory perception.The psychic was presented with 200 cards face down and asked to determine if each card were one of five symbols: a star,a cross,a circle,a square,or three wavy lines.The psychic was correct in 50 cases.Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial.Assume the 200 trials can be treated as a simple random sample from the population of all guesses the psychic would make in his lifetime.What do we know about the value of the P-value for the hypothesis test? (Note: Use the large-sample z statistic. )

A simple random sample of 200 students is selected from a large university.In this sample,there are 35 minority students.A simple random sample of 80 students is selected from the community college in the same town.In this sample,there are 28 minority students.What is the standard error of the difference in sample proportions of minority students?

A quality manager in a small manufacturing company wants to estimate the proportion of items produced by a very specialized process that fail to meet a customer's specification.Because it is very expensive to determine if an item produced by the process meets the specification,only a very small number of items can be tested.A random sample of 15 items was selected,and in the sample three of them failed.What is the standard error of the plus four estimate of the true proportion p?

A manufacturer receives parts from two suppliers.A simple random sample of 400 parts from Supplier 1 finds 20 defective.A simple random sample of 200 parts from Supplier 2 finds 20 defective.Let p₁ and p₂ be the proportion of all parts from Supplier 1 and Supplier 2,respectively,that are defective.Is the difference statistically significant at the 5% significance level? To determine this,test the hypotheses H₀: p₁ = p₂ versus H_a: p₁  p₂.

A simple random sample of 100 bags of tortilla chips produced at Plant A is selected every hour for quality control.In the current sample,18 bags had more chips (measured in weight)than the labeled quantity.This sample information is being compared to data collected at Plant B.In the current sample of 100 bags of chips from Plant B,12 bags had more chips than the labeled quantity.What is the relative risk that a bag of tortilla chips produced at Plant A had more chips than the labeled quantity as compared to Plant B?

After a college football team once again lost a game to their archrival,the alumni association conducted a survey to see if alumni were in favor of firing the coach.A simple random sample of 100 alumni from the population of all living alumni was taken.Sixty-four of the alumni in the sample were in favor of firing the coach.Let p represent the proportion of all living alumni who favored firing the coach.Suppose the alumni association wished to see if the majority of alumni are in favor of firing the coach.To do this they test the hypotheses H₀: p = 0.50 versus H_a: p > 0.50.What is the P-value for this hypothesis test?

The Information Technology Department at a large university wishes to estimate the proportion of students living in the dormitories,p,who own a computer with a 95% confidence interval.What is the minimum required sample size the IT Department should use to estimate the proportion p with a margin of error no larger than 3 percentage points?

A newspaper is conducting a statewide survey concerning the race for governor.The newspaper will take a simple random sample of n registered voters and determine X = the number of voters that will vote for the Democratic candidate.Is there evidence that a clear majority of the population will vote for the Democratic candidate? To answer this,they will test the hypotheses H₀: p = 0.50 versus H_a: p > 0.50.If n = 120 and X = 64,what is the P-value for this hypothesis test?

Suppose that =.5, = .2,n₁ = 20,and n₂ = 30.What is the number of successes,X₂,based on p₂?

A noted psychic was tested for extrasensory perception.The psychic was presented with 200 cards face down and asked to determine if each card were one of five symbols: a star,a cross,a circle,a square,or three wavy lines.The psychic was correct in 50 cases.Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial.Assume the 200 trials can be treated as a simple random sample from the population of all guesses the psychic would make in his lifetime.Based on the results of the test,what is a 95% confidence interval for p?

Suppose you are conducting a study to determine if women are better drivers than men.You send your survey to over 1000 students on your campus and 5 students respond that women are better drivers and 4 students respond that women are not better drivers.Which of the methods below should you use to build a confidence interval?

It is estimated that 780,000 surgical site infections (SSIs)occur each year.SSIs are the second most common type of healthcare-associated infections in U.S.hospitals and account for an extra $3.5 to $10 billion in healthcare costs per year.The national SSIs rate is 1.9%.A Georgetown medical office was interested in determining if their SSI rate were smaller than the national average.Out of a sample of 277 patients in their study,only one infection occurred.What is the sample proportion ?

Suppose that =.5, = .2,n₁ = 20,and n₂ = 30.What is the number of successes,X₁,based on p₁?

A simple random sample of 100 of a certain popular model car in 2003 found that 20 had a certain minor defect in the brakes.A simple random sample of 400 of this model car in 2004 found that 50 had the minor defect in the brakes.Let p₁ and p₂ be the proportion of all cars of this model in 2003 and 2004,respectively,that actually have the defect.What is a 90% confidence interval for p₁ - p₂?