Confidence Intervals for Difference in Proportions
Confidence Intervals for Difference in Means
Old Topics
100
UFC is conducting a study to determine if the US has a higher proportion of winning fighters compared to Europe. They create a 95% CI for the proportion of winning US fighters - proportion of winning Europe fighters. The interval they obtain is (0.23, 0.4). Interpret this interval in context.
We are 95% confident that the true difference in proportion of winning fighters between the US and Europe is between 0.23 and 0.4.
100
Two racers compare their average race time over 50 races. They then find the 95% CI for the difference in means race times (in minutes) between the two racers. They find this interval to be (-1.2, 2). Interpret the CI in context.
We are 95% confident that the true difference in mean race time between the two racers is between -1.2 minutes and 2 minutes.
100
What type of data can be visualized with dot plots, histograms, etc.?
Quantitative Data
200
A company is testing the enjoyment of two of its products, Product A and Product B. They construct a 95% confidence interval for the difference in proportion of customers who recommend Product A vs. Product B. The interval they find is (-0.1, 0.05), what can be concluded from this interval?
Since 0 falls inside this interval, there is not significant evidence that the proportion of customers who would recommend Product A is different from the proportion of customers who would recommend Product B.
200
An archeologist is comparing the average number of fossils found a day at two dig sites. They construct a 95% CI for the difference in mean number of fossils found per day between dig sites (Using Dig Site 1 - Dig Site 2). The interval they construct is (-3.5, -1.2). Test the claim that Dig Site 1 has the same mean number of fossils found per day as Dig Site 2.
No, since the entire interval is negative we would have convincing evidence that the mean number of fossils found per day at Dig Site 1 is less than Dig Site 2.
200
A poll is given to customers on whether they would buy a product based on one of three separate commercial versions being shown. The data is recorded in the two-way table below.
Find:
P( Likely Buy | Version 1)
P(Version 2| Unsure or Unlikely)
P( Likely Buy | Version 1)=2/3
P(Version 2| Unsure or Unlikely)=10/81
300
A company is hiring and wants to determine the most effective way to obtain qualified candidates. They specifically want to compare Indeed and Personal Recommendation. They take a random sample of 50 Indeed candidates and find 10 are qualified for the job, whereas they take a random sample of 30 Personal Recommendation candidates and find 15 are qualified. Find a 95% CI for the difference (prop. of qualified Indeed candidate)-(prop. of qualified Personal Recommendation candidate)
(-0.509, -0.090)
300
An ice cream truck owner wants to compare the mean number of ice cream bars sold on Wednesdays vs. Saturdays. Assume this ice cream truck has been around for over 15 years. They select a random sample of 50 Wednesdays and 45 Saturdays. They find that they sell an average of 60 bars on Wednesdays with a standard deviation of 5 bars, whereas they sell an average of 110 bars with a standard deviation of 20 bars on Saturday. Construct a 95% CI for the difference in mean ice cream bars sold on Wednesday vs. Saturday. (Use: Wed-Sat)
(-56.16, -43.84)
300
What type of sampling splits the population and then samples randomly in these split subpopulations?
Stratified Sampling
400
A sports analyst for the NBA is comparing the Laker's and Golden State Warrior's proportion of games with more than 30 three point attempts over the last 10 years. They take a random sample of 60 games played in the last 10 years from each team. They find 60% of the games in the Lakers sample and 70% of the Golden State Warriors sample had more than 30 three point attempts. Construct a 95% CI for the difference in proportion of games with more than 30 three point attempts. (Use Lakers - GSW)
(-0.271, 0.071)
400
A gambler is comparing how much money he makes on the first spin of two different slot machine types. He randomly selects 40 penny slots and 60 quarter slots and makes a bet of $0.25 on both. He finds the mean money made on the first spin of the penny slots from his sample is -$0.10 with a standard deviation of $0.03, and the mean money made on the first spin of the quarter slots to be $0.04 with a standard deviation of $0.10. Find a 95% CI for the difference in mean money made on the first spin between penny and quarter slots. (Use: Penny - Quarter).
(-0.167, -0.113)
400
Oaks Christian is determining what the average commute distance for students is. They take a random sample of 200 students. Would we be able to use the sample standard deviation of sigma/sqrt(200 ? Explain your reasoning.
No, despite having a random sample and meeting the CLT condition, they have too large a sample which is greater than 10% of the Oaks Christian population.
500
Oaks Christian School is trying to address an attendance issue among seniors. They take a random sample of 20 seniors and 25 juniors on a given day. They find that 50% of the seniors are absent and 40% of the juniors are absent. Construct and Interpret a 95% CI for the difference in (Proportion of seniors who are absent)-(Proportion of juniors who are absent). Can we conclude that the proportion of seniors absent on this day is higher than the proportion of juniors absent? Explain.
(-0.192, 0.392)
We are 95% confident that the true difference in proportion of seniors who are absent on this day and juniors who are absent on this day is between -0.192 and 0.392.
Since 0 falls inside our interval, we cannot conclude that there is a difference in the proportion of absent seniors and absent juniors.
500
A coffee shop is comparing how much customers spend on average when they sit down vs. when they drive thru for the previous year. They randomly select 90 orders from sit down customers and randomly select 45 orders from drive thru customers. They find the average of these sit down orders to be $9.42 with a standard deviation of $1.45, as well as the average of these drive thru orders to be $13.56 with a standard deviation of $2.10. Find and interpret the 95% CI for the difference in mean amount spent per order between sit down and drive thru customers. (Use: Sit - Drive). Test the claim that drive thru customers typically spend more than sit down.
(-4.84, -3.44)
We are 95% confident that the true difference in mean amount spent by sit down customers vs. drive thru customers is between -$4.84 and -$3.44.
Since all of the CI is negative, there is convincing evidence that drive thru customers do spend more per order on average than sit down customers.
500
The true mean listens for country songs is 22,000 with a standard deviation of 1,150, whereas the true mean listens for rap music is 37,000 with a standard deviation of 3,200. Find the probability that a random sample of size 40 rap songs have at least 16,000 more mean listens than a random sample of 35 country songs.
P(X>=16000)=0.0325