Exam 19: Comparing Two Proportions

A new manager started a program on employee sick leave.The manager later compares the sick leave rate prior to the program starting (65 out of 250 employees were sick)to the sick leave rate one month into the program (55 out of 253 employees were sick).

Use the given sample data to test the claim that p1 < p2.Use a significance level of 0.10. $\text { Sample } 1$ $\text { Sample } 2$ n1 = 462 n2 = 380 x1 = 84 x2 = 95

Suppose the proportion of women who watch CBC News is $\mathrm { P } _ { \mathrm { W } }$ and the proportion of men who watch CBC News is $\mathrm { Pm }$ .The survey found a 95% confidence interval for $p _ { W } - p _ { m }$ is $( - 0.06,0.25 )$ Give an interpretation of this confidence interval.

A marketing survey involves product recognition in Ontario and British Columbia.Suppose the proportion of Ontario residents who recognized a product is $\mathrm { P } 1$ and the proportion of British Columbia residents who recognized the product is $\mathrm { P } 2$ .The survey found a 98% confidence interval for $p _ { 1 } - p _ { 2 }$ is $( - 0.023 , - 0.019 )$ Give an interpretation of this confidence interval.

A poll reported that 3 out of 50 first year university students surveyed did not have jobs,while 7 out of 50 second year university students did not have jobs during the academic year.

In words,how would you find the standard deviation of the difference between two independent random variables?

In City A,7420 people are unemployed in a random sample of 143000 people.In City B,7642 people are unemployed in a random sample of 142000 people.

An Ontario study examined the effect of day care on behavior in toddlers.Randomly selected parents who had a toddler in full-time day care were asked if their child had behavioral problems.The researchers found that among 987 parents surveyed,212 said their child had behavioral problems.Among 349 randomly selected parents with a toddler at home,17 reported that their child had behavioral problems.

Suppose the proportion of women who follow a regular exercise program is $\mathrm { P } _ { \mathrm { W } }$ and the proportion of men who follow a regular exercise program is $\mathrm { Pm }$ .A study found a 98% confidence interval for $p _ { W } - p _ { m }$ is $( - 0.022,0.111 )$ Does this study provide strong evidence of a gender-based difference in exercise? Explain.

7 of 8,500 people vaccinated against a certain disease later developed the disease.18 of 10,000 people vaccinated with a placebo later developed the disease.Test the claim that the vaccine is effective in lowering the incidence of the disease.Use a significance level of 0.02.

In a random sample of 360 women,65% watched CBC News.In a random sample of 220 men,60% watched CBC News.Test the claim that the proportion of women who watch CBC News is higher than the proportion of men who watch CBC News.Use a significance level of 0.05.

A poll checking on the level of public support for a tax increase reported that 19% of the respondents were in favour.The pollsters reported a sampling error of ±5%.When the responses were broken down by gender,support was 3% higher among women than men.The pollsters said the margin of error for this difference was ±6%.Why is the margin of error larger for the difference in support between the genders than for the overall level of support?

n1 = 50 n2 = 75 X1 = 20 x2 = 15

A university is studying the proportion of students retained in various years.Suppose the proportion of students retained in 2012 is $\text { P12 }$ and the proportion of students retained in 2013 is $\mathrm { P } 13$ .A study found a 90% confidence interval for $\text { P12 }$ - $\mathrm { P } 13$ is (-0.0398,0.0262).Give an interpretation of this confidence interval.

A survey of randomly selected university students found that 83 of the 110 first year students and 87 of the 120 second year students surveyed had purchased used textbooks in the past year.Construct a 98% confidence interval for the difference in the proportions of first and second year university students who purchased used textbooks.

A poll reported that 30% of 60 Canadians between the ages of 25 and 29 had started saving money for retirement.Of the 40 Canadians surveyed between the ages of 21 and 24,25% had started saving for retirement.

Suppose that Statistics Canada wants to compare the current unemployment rate between Montreal and Toronto.Let $\mathrm { PM }$ and $\mathrm { PT }$ represent the current unemployment rates in Montreal and the current A study found a 90% confidence interval for $\mathrm { PM }$ - $\mathrm { PT }$ is (-0.0457,0.0424).Give an interpretation of this confidence interval.

A survey of randomly chosen adults found that 30 of the 64 women and 49 of the 74 men follow regular exercise programs.Construct a 95% confidence interval for the difference in the proportions of women and men who have regular exercise programs.

A study investigated the smoking status of people in two age groups.A random sample of 100 people aged 20-24 revealed 24 who are smokers.A random sample of 120 people aged 25-29 revealed 15 who are smokers.

You wish to construct a 95% confidence interval to compare two proportions.If we wish to obtain a margin of error of at most 0.03,what sample size should we take from each group? Assume equal sample sizes.