Understanding Two-Sample T-Tests: Methods and Examples
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Dec 10, 2024
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Two-sample t-testRecap/review on one sample z statistic and t statisticChapter 7Two-sample t statistic
I took a sample of the grade-point averages for students in my class. For 25 students, the standard deviation of grade points was 0.65 and the mean was 2.89. The standard error for the sample wasA. 0.578.B. 0.026.C. 0.130.
To construct a 95% confidence interval with a sample size of 15, the correct t* to use isA. 2.145.B. 1.761.C. 1.645.
Below are normal quantile plots for two data sets. For which set(s) is/are inference using the tdistribution valid?a.b.c. both (a) and (b)d. neither (a) nor (b)
The manager of an automobile dealership is considering a new bonus plan to increase sales. Currently, the mean sales rate per salesperson is five automobiles per month. The correct set of hypotheses to test the effect of the bonus plan isA. H0: = 5, Ha: > 5.B. H0: > 5, Ha: = 5.C. H0: = 5, Ha: > 5.D. H0: = 5, Ha: = 5.
Problems taken from:STAT 301 Exam 2Spring 2019•You should have z and t tables with you•You should have a calculator•You should have your reference sheet started with the formulas to refer to•If you just sit and watch me solve the problems, that isn’t going to help you do well on this exam
USE THE FOLLOWING TO ANSWER QUESTIONS 1 – 4: What is the mean amount of water an adult Asian elephant drinks daily? Researchers selected a sample of 112 adult Asian elephants from two National Parks in India. Data was recorded on the daily amount of water drunk by each elephant in the sample. Based on the data, the mean was 29.6 gallons and the standard deviation was 1.2 gallons (Write down letter answers on the lines next to the question’s number).________1. (3 points) In this problem:A.µ = 29.6 and s = 1.2.B.= 29.6 and s = 1.2.C.= 29.6 and σ = 1.2.D.µ = 29.6 and σ = 1.2.________2. (3 point) Researchers decided to calculate a confidence interval (at the 99% confidence level) to estimate the mean amount of water an adult Asian elephant drinks daily. E.The confidence interval will be centered at the population mean.F.The confidence interval will be centered at the sample mean. G.The confidence interval will provide information of the exact range of values the population mean can take. H.The confidence interval at the 95% confidence level will be bigger than at the 99% confidence level.
USE THE FOLLOWING TO ANSWER QUESTIONS 1 – 4: What is the mean amount of water an adult Asian elephant drinks daily? Researchers selected a sample of 112 adult Asian elephants from two National Parks in India. Data was recorded on the daily amount of water drunk by each elephant in the sample. Based on the data, the mean was 29.6 gallons and the standard deviation was 1.2 gallons (Write down letter answers on the lines next to the question’s number).________3. (3 point) If the researchers decide to calculate a confidence interval at the 99% confidence level, the value of z* or t* that will be needed is approximately:2.332.6262.5762.603________ 4. (3 points) If the researchers decide to calculate a confidence interval at the 99% confidence level, the margin of error will be: 0.2983.0910.025
5. Five years ago college graduates left school with an average of $28,000 in student loans. Is there evidence that the average student loan at graduation is greater now than it was ten years ago?In order to answer this question, data was collected from a sample of 81 college students who graduated within the past year. The average student loan for this sample was $29,345 with a standard deviation of $8,567. Follow the steps to calculate the corresponding significance test that will give an answer to the above question, at the 1% significance level. Show all your work.a) (4 points) Write the null and alternative hypotheses. Make sure to use the correct notation (1 point for using the correct notation, 2 points for correct signs in H0and Ha, 1 point for correct value of the null).H0:Ha b) (3 points) Calculate the test statistic. Make sure to state whether this is a Z, T, or F statistic. Make sure you check all the information that is known for this problem as well as the number of samples and number of variables measured. Write down the equation for calculating itand substitute the problem’s values. Give your answer to 2 decimal places. (1 point for correct test statistic, 1 point for correct equation, 1 point for correct values in the equation).
5. Five years ago college graduates left school with an average of $28,000 in student loans. Is there evidence that the average student loan at graduation is greater now than it was ten years ago?In order to answer this question, data was collected from a sample of 81 college students who graduated within the past year. The average student loan for this sample was $29,345 with a standard deviation of $8,567. Follow the steps to calculate the corresponding significance test that will give an answer to the above question, at the 1% significance level. Show all your work.c) (2 point) Write the probability statement corresponding to the test statistic. d) (3 points) If the curve below corresponds to the density curve of the test statistic, indicate the value at the center for the test statistic (1 point), indicate the location of the test statistic for this problem (1 point) and shade the area corresponding to the P-value (1 point).
5. Five years ago college graduates left school with an average of $28,000 in student loans. Is there evidence that the average student loan at graduation is greater now than it was ten years ago?In order to answer this question, data was collected from a sample of 81 college students who graduated within the past year. The average student loan for this sample was $29,345 with a standard deviation of $8,567. Follow the steps to calculate the corresponding significance test that will give an answer to the above question, at the 1% significance level. Show all your work.e) (2 point) Find the P-value (or interval within which the P-value would be) from one of the provided tables. f) (4 points) Write the conclusion to the hypothesis test in terms of the problem. Be sure to mention whether the conclusion applies to the population or the sample.
6. (3 points) (Write your answer on the line next to the question’s number). A confidence interval can be used to assess whether or not there is significant evidence supporting an alternative hypothesis,A.If the alternative hypothesis is one-sided.B.If the alternative hypothesis is double-sided.C.If the significance level plus the confidence level add up to 100%.D.If the significance level plus the confidence level add up to 100%, independently of the type of alternative hypothesis.E.Only if A and C are both verified. F.Only if B and C are both verified.
Work on the following problem.PopulationAll STAT students𝜇SRS1001 STAT301 StudentsWe are interested in the average final grade for elementary statistics students at universities in the USA.1)C.I. What is the interval around the measured mean within which I will be 95% confident of finding the value of the average final grade for the entire population (µ)? 2) Significance test There is a claim that the average final grade is 80. Is there any evidence of the contrary based on the collected data?
Recap on one sample z and t statisticsIf we know Confidence IntervalHypothesis testing=If we know s (sample)Confidence IntervalHypothesis testing=FIND PROBABILITY FROM TABLE
CI and Hypothesis TestingOne sample z- test•WE KNOW •Parameter of interest: •Measure: CI•Find z from t-tableHT•Find p from z-tableOne sample t-test•WE DON’T KNOW •Parameter of interest: •Measure: CI•Find t from t-tableHT•Find p from t-tableMatched pairs•WE DON’T KNOW •Parameter of interest: •Measure: CI•Find t from t-tableHT•Find p from t-table
What if we want to compare 2 populations?Compare the responses of 2 groups (independent from each other). Each group is considered a sample from a different population.Examples: male vs female, control group vs non-control, one section of STAT301 vs. another section of STAT301 Population Mean STD Sample Mean STD1 μ1 σ11 1s12 μ2σ22 2s2
Inference in a 2-sample t-test(2 samples, we measure the same response variable for both)C.I.± tSignificance testHo: µ1=µ2or µ1-µ2=0Ha: µ1>µ2or µ1-µ2>0 µ1<µ2or µ1-µ2<0 µ1≠µ2or µ1-µ2≠0If possible choose 1st as the sample with the largest sample mean so you have positive number. d.f.=n-1 where n=min(, )d.f.=n-1 where n=min(, )µ1-µ2 under the null hypothesis. SinceHo: µ1-µ2=0, that would be the value in this equationt=
CI and Hypothesis Testing T-TestsMatched pairs•WE DON’T KNOW •Parameter of interest: •Measure: CIFind t from t-tableHTFind p from t-tableTwo sample t-test•WE DON’T KNOW , •Parameter of interest: - •Measure: CIFind t from t-tableHTFind p from t-table
2-sample comparison of means example using t- tableA study is carried out to see if afternoon STAT301 students get more sleep than morning STAT301 students.CEMThis Photoby Unknown Author is licensed under CC BY-NC-ND
2-sample comparison of means example using t-tableAM Class•N = 61PM Class Combined•N = 116
2 sample comparison of means example using t-tableGroup 1: AM Class, N = 61, Group 2: PM Class, N = 116, H0: u1 = u2Ha: u1 < u2
2-sample comparison of means example using SPSS A study is carried out to compare the time/week spent doing housework between male and female college students.L.C.
2-sample comparison of means example using SPSSData for women :Data for men :Test whether the population mean weekly time spent on housework by men is different than the population mean weekly time spent on housework by women. 205821761591111371602572331092411721481711441461941922658556666435957728282436136386090446755
2-sample comparison of means example using SPSSPooled d.f.As estimated by the software (see text book)Use lower row results. Most conservativeT calculatedTwo-sided P-value95% C.I.Notice what SPSS calls this
2-sample comparison of means example using SPSS
2-sample comparison of means example using SPSS (one sided test)Most studies have found that the population mean of time/week spent on housework by male college students is lower than the population mean time/week spent on housework for a comparable group for female college students. Perform this hypothesis test.
2-sample comparison of means example using SPSS
2-sample comparison of means example using SPSSGive a 95% confidence interval for the difference in the population mean time/week spent on housework by male and female college students.
OK to use t procedures?1-sample and matched pairs: use n2-sample comparison of means: use n=n1+n2Sample sizeUse t proceduresn < 15if your data looks like a Normal distribution15 ≤ n < 40except in the presence of outliers or strong skewnessn ≥ 40except in the presence of outliers, ok even if data are skewed
Summary of inference techniquesHypothesis test vs. confidence interval?CI if you are looking for an estimate for your population mean.HT if you are checking your population mean against a number or another population mean.
Summary of test statisticsz vs. t for inference with the mean?•zif you know the population standard deviation, σ•tif you only know the sample standard deviation (more realistic case), sMatched pairs vs. 2-sample comparison of means?•Matched pairs : We have one sample and we record two response variables for each unit.•2-sample comparison of : We have two samples and we record the same response variable for all units in both samples.
Summary of t-test outputs from SPSS and Ho statementsHo:µ=µ0Ho:µdiff=µdiff,0Ho:µ1=µ2or µ1-µ2=0