Sampling Proportions
We want to estimate the percentage of seniors at Whitman who drive to school. We take a SRS of 25 seniors and find 15 who drive (15/25 = 0.6). If we take a different SRS of 25 seniors will the sample proportion still be 0.60?
What is... Not necessarily. (The sample proportions from repeated samples will vary.)
Suppose an SRS of 200 Americans finds 95 that support a new law. Lets find a 95% confidence interval for the true percentage.
What is... I am 95% confident that the true proportion of Americans that support this new law is between 40.58% and 54.42%.
A principal in MCPS claims that Whitman Ivy League acceptances are 16% while Dr. Dodd claims that it is definitely greater than 16%. Dr. Dodd takes a SRS of 120 Whitman students finding 30 that were accepted to Ivy Leagues. Testing at the 5% level, is Dr. Dodd right?
What is... P(z > 2.6893) = 0.0036. At the 5% level this is significant. Based on the data Dr. Dodd can confidently say that the percentage of Whitman students who get accepted into Ivy League schools is greater than 16%.
You're on a plane that's about to crash and there's 2 parachutes: the one on you and the second one next to you. You have to choose which one to wear and which one to leave. What is the null hypothesis in this scenario?
What is...The parachute on your back is safe to use.
A SRS of 70 coal miners found that 50 were in a union while a SRS of 80 construction workers found that 60 were in a union. Find and interpret a 95% CI for the true difference in proportions between the two types of workers.
What is... I am 95% confident that the true percentage of unionized construction workers is between 10.64% lower and 17.8% higher than the percentage of coal miners.
Suppose 10% of the population of members at a fitness club is left handed (Population = 2000) We take a SRS of 120 members. What is the probability that our sample proportion is greater than 0.15?
What is... P(p>.15) = 0.0344
Suppose 72 of 100 randomly selected cookies have M&Ms in them. How many cookies should be in a new sample to be 95% confident of the true percentage of M&M cookies accurate to within +2%.
What is... 1936 cookies.
Joe Biden said if he ran for president in 2024 he claims that 45% of MD voters would vote for him. A member of his family thinks the true percentage is higher, so we take a SRS of 300 MD voters, finding 150 who would vote for him. Testing at the 5% level is there evidence to suggest that Biden's claim was too low?
What is... P(z > 1.7408) = 0.0409. At the 5% level this is significant. Based on this data we can conclude that the true percentage of Maryland voters who would vote for Biden in 2024 is greater than 45%.
Suppose the County wants to estimate the percent of students who take the bus to school. If they believe that the percent is less than 20% then they will eliminate the bus service. If they believe that it is greater than 20% they will add more buses to the system. What is the Type II error in this case?
What is... The county concludes the percentage is greater than 20% and add buses to the system when in reality less than 20% of students take the bus.
A SRS of 120 History teachers found that 30 admitted to yelling at students in the classroom while a SRS of 100 English teachers found that 40 admitted to yelling at their students. Assuming that these teachers aren't lying, test at the 5% level to see if it looks like History teachers cuss less than English teachers.
What is... If the true difference between the proportion of History teachers that cuss and the English teachers that cuss is really 0 then the probability of getting a difference of 15% or more extreme is .87%. Based on this data we can conclude that the difference in proportions of cussing between History and English teachers is not 0 and History teachers cuss less often.
Suppose 75% of all dog owners take their dogs to the vet for regular checkups. We take a SRS of 150 dog owners. What is the probability that less than 70% of them take their pets to the vet regularly?
What is P(p < 0.7) = 0.0793
The Governor claims that half the Maryland population wants to legalize marijuana. A SRS of 250 Maryland residents finds 75 who support the legalization. Does a 95% CI support the governors claim.
What is... No, based on the confidence interval the Governor's claim is not supported.
A recent SRS of 85 Walmart customers found 60 who think Walmart is the best store chain in the country. The Walmart country advertises that 75% of its customers feel that way. Based on this data is Walmart's claim true?
What is P(z < -.9393) = 0.1738. At 5% level this is not significant. Based on this data we cannot conclude that the true percentage of Walmart shoppers that think its the best is less than 75%.
Suppose the County wants to estimate the percent of students who take the bus to school. If they believe that the percent is less than 20% then they will eliminate the bus service. If they believe that it is greater than 20% they will add more buses to the system. What is the Type I error in this case?
What is... The county concludes that the percent is lower than 20% and eliminates the bus service when more tahn 20% of students take the bus to school.
Suppose a random sample of Whitman Students found 75 out of 100 are in favor of not having school the day after halloween. Meanwhile, a random sample of 120 Wootton students found 85 who were in favor of the same. Find a 90% CI for the true difference in proportions for the two schools.
If it is known that 70% of Maryland drivers exceed the speed limit on highways. We take a SRS of 50 Maryland drivers. What is the probability that the sampling distribution shows a proportion of less than 60%
What is... P(p < 0.6) = 0.0618
How many Popeyes employees do we have to survey to be 90% confident of estimating the true percentage of blue eyed people accurate to +3%.
What is 752 employees.
60% of the American population has brown hair and 22% is black. A teacher takes a SRS of 50 Whitman students and finds that 34 of them are brunette. Can he conclude that Whitman students hair is different to the American population.
What is... P(z>1.15) = 0.248. At 5% level this is not significant. Based on this data we cannot conclude that the Whitman students hair is different to the American population.
Education experts want to know how effective the current SAT Reading format is. In the previous version, 69% of students in a school district performed proficiently on previous SAT format. Experts want to know if results are better with the current SAT What is the Type I error in this scenario?
What is... claiming that in the current SAT format more than 69% students performed proficiently when they did not.
Suppose a random sample of 150 dog owners found that 45 of them have an electrical fence system at their homes. A random sample of 125 cat owners found that 27 of them have an electrical fence system at their homes. Testing at the 10% level is there evidence to suggest that a higher percentage of dog owners than dog owners have electrical fences.
What is... If the difference between dog and cat owners who have electrical fences really is 0 then the likelihood of repeating the observed difference of 8.4% or more extreme would be 5.73%. At a significance level of 10% we reject the null hypothesis and conclude that the percentage of dog owners that have electrical fences is greater than the percentage of cat owners.
Suppose that Red Skittles are supposed to make up 30% of the candies sold. Consider a randomly selected bag of 250 Skittles. What is the probability that this random bag of skittles contains at least 25% Red skittles?
What is... P(p > 0.25) = 0.952
145 of 180 randomly selected US residents say they recycle regularly. The goal is to have 90% of all residents recycling. Would a 95% confidence interval for the SRS make it look like the goal has been been achieved.
What is ... No, based on the confidence interval the goal has not been reached.
63% of engineering applicants were accepted into engineering schools. A program that trains engineering students claims that 165 of 240 students in their program made it into engineering schools. Is the program doing better than the national average?
What is... No at 5% it is not significant.
Education experts want to know how effective the current SAT Reading format is. In the previous version, 69% of students in a school district performed proficiently on previous SAT format. Experts want to know if results are better with the current SAT. What is the Type II error in this scenario?
What is... They concluded that in the current SAT format there is no increase in percentage of students who performed proficiently when in reality there was an increase.
72 of 100 Popeyes eaters prefer the spicy sauce on their sandwhiches. 85 out of 110 Chick-Fil-A eaters prefer the Chick-fil-a sauce. Test at the 10% level to see if a higher proportion of Chick-Fil-A eaters prefer the Chick-Fil-A sauce.
What is... If there really was no difference in the proportion of Popeyes eaters who prefer the spicy sauce and Chick-Fil-A eaters who prefer the Chick-Fil-A sauce then the likelihood of getting a difference of 0.053 or more extreme is 18.98%. Testing at the 10% level this is not significant. We cannot reject the null and we cannot conclude that there is a difference in the proportions of what sauce Popeyes and Chick-Fil-A eaters prefer.