This corequisite support activity reviews sampling with replacement, sampling without replacement, and confidence intervals for a population mean using the tDistribution. Understanding the difference between sampling with and without replacement is important because this lesson introduces a new method for calculating a confidence interval for a population mean that is based on sampling with replacement.
Sampling With and Without Replacement
The usual way to select a random sample from a population is using sampling without replacement. This means that once an individual from the population is selected for the sample and data are recorded for that individual, they are not considered again when making additional selections from the population for that sample. In sampling without replacement, no individual is included in the sample more than once. Sampling with replacement is different in that after an individual is selected for the sample and data are recorded for that individual, they are “replaced” (put back into the population) before the next selection is made. This means that it is possible that the same individual might be selected again and their data would then be included more than once in the sample.
Questions 1–4
Francisco lives in a college dorm that encourages students to engage in community service. Francisco plans to take a sample of 20 students from the 200 students living in his dorm to learn about the average number of hours per month spent doing community service. Here is how he plans to select his sample:
•Obtain a list of the 200 students who live in the dorm.
•Number the students on the list from 1 to 200.
•Use a random number generator to obtain 20 random numbers between 1 and 200. If any numbers are repeated in the list of 20 random numbers, he will ignore the repeats and generate additional random numbers until he has 20 numbers that are all different.
•Create a list of 20 students by using the random numbers to identify which students on his numbered list of 200 students will be included in the sample.
•Contact each of the students selected for the sample and record the number of hours per month that they report spending doing community service.
Question 1
1) Is Francisco selecting his sample with replacement or without replacement? What part of his plan supports your answer?
Question 2
2) If Francisco wanted to select his sample with replacement, how should he modify his plan?
Question 3
3) If Francisco selects his sample of 20 studentswith replacement, is he guaranteed tohave at least one student included more than once in his sample? Explain.
Question 4
4) If Francisco is going to select a sample of size 20 with replacement, would he be more likely to include a student more than once if his dorm only had 50 students rather than 200 students? Explain.
The t Confidence Interval for a Population Mean
Recall that if certain conditions are met, a one-sample t confidence interval can be used to estimate a population mean. The conditions for the one-sample t confidence interval are:
•The sample is a random sample from the population of interest.
•The sample size is at least 30 or the population distribution is approximately normal.
Go to https://dcmathpathways.shinyapps.io/Inference_mean/.•Choose the “Cell Phone Reaction Times” dataset from the dropdown menu.
•For “Type of Inference,” select “Confidence Interval” and set the confidence level to 95%. Make sure that “Interval” is selected where it asks “Interval, Lower or Upper Bound?” You may recall that the “Cell Phone Reaction Times” dataset consists of differences in reaction times (in milliseconds) recorded for a sample of people both when they were using cell phones and when they were not using cell phones.Use the output on the right-hand side of the screen to answer the following questions.
Question 5
5) The differences were calculated as reaction time with cell phone minus reaction time without cell phone. What does it meanfor a difference to be positive?
Question 6
6) How many people were in the sample?
Question 7
7) Assuming that the sample is representative and can be regarded as a random sample of adults, are the conditions necessary for the one-sample t confidence interval met? Explain.
Question 8
8) What is a 95% confidence interval for the mean difference in reaction time?
Question 9
9) Interpret the 95% confidence interval in Question 8.
Go to https://dcmathpathways.shinyapps.io/EDA_quantitative/and choose the “CO2 Emissions of EU Countries” dataset from the dropdown menu.(It is near the bottom, so you will probably need to scroll down to find it.)Use the output on the right-hand side of the screen to answer the following questions.
Question 10
10) How many countries were in the sample?
Question 11
11) This dataset consists of the CO2 emissions (in metric tons) per person for a sample of countries in the European Union(EU). Based on the graphs of the dataset, would it be appropriate to use the data from this sample to construct a t confidence interval for the mean CO2 emissions per person for EU countries? Explain.
