T-Test

Question 1 (40 points)

According to the National Association of Colleges and Employers, the average salary for a new college graduate in 2019 was $45,000. A small academic program wants to know if the average salary of its graduates is below the national mean value in 2019. If it is below the national mean value, they want to take measures to increase the starting salary of their graduates. The starting salary of 36 recent graduates of this program is stored in an R vector called “college_salary”. Copy the following text, paste it in an R studio code chunk, and run the code chunk. Afterward, you will have the vector available in your R project.

college_salary= c (45600, 51400, 35600, 42892, 48180, 51000, 39900, 40000, 48100, 42033 ,41072, 42898, 42010, 51200, 43260, 42901, 40296, 44350, 43200, 47512, 45590, 50050, 41110, 42194, 44643, 43600, 40600, 50961, 39254, 42400, 52000, 46100, 50533, 42026, 41236, 44239)

(25 points) Conduct the corresponding hypothesis test. Report the following as part of your answer: The hypotheses (Ho and Ha). The P value. Your conclusion.

Ho: average (mean) salary of graduates >= $45,000

Ha: average (mean) salary of graduates < $ 45,000

college_salary= c (45600, 51400, 35600, 42892, 48180, 51000, 39900, 40000, 48100, 42033 ,41072, 42898, 42010, 51200, 43260, 42901, 40296, 44350, 43200, 47512, 45590, 50050, 41110, 42194, 44643, 43600, 40600, 50961, 39254, 42400, 52000, 46100, 50533, 42026, 41236, 44239)

college_salary

##  [1] 45600 51400 35600 42892 48180 51000 39900 40000 48100 42033 41072 42898
## [13] 42010 51200 43260 42901 40296 44350 43200 47512 45590 50050 41110 42194
## [25] 44643 43600 40600 50961 39254 42400 52000 46100 50533 42026 41236 44239

(10 points) Show the computations needed to get the value of the test statistic that you obtained in the R output from part a. In other words, you need to show how that value was obtained.

mean(college_salary)

## [1] 44442.78

sd(college_salary)

## [1] 4124.048

Just because the sample mean is less than $45,000 it does not mean that, at the population level (all the salaries), the average is less than $45,000. We need to figure out is $44,442 is sufficiently far away from $45,000. The test statistic and the p-value will tell us.

(5 points) What type of statistical error could you potentially be making after your decision in part a? Justify.

t.test(college_salary, alternative = c("less"), mu=45000)

## 
##  One Sample t-test
## 
## data:  college_salary
## t = -0.81069, df = 35, p-value = 0.2115
## alternative hypothesis: true mean is less than 45000
## 95 percent confidence interval:
##      -Inf 45604.09
## sample estimates:
## mean of x 
##  44442.78

Conclusion: p-value = 0.2115 > alpha (0.05); therefore, we fail reject Ho.

Question 2 (60 points) Researchers at a Zoo wish to test the efficacy of a new program intended to reduce the length of labor in female pumas (i.e., mountain lions). From historical records, researchers know that it takes an average of 60 minutes for a female puma to give birth to a litter. The new program was tested with 25 puma mothers that gave birth. The data of the length of labor (in minutes) for these 25 puma mothers is shown below.

puma_labor= c(56, 51, 62, 58, 59, 59, 56, 56, 54, 47, 51, 64, 48, 59, 59, 53, 61, 55, 57, 63, 55, 56, 62, 57, 55)

puma_labor= c(56, 51, 62, 58, 59, 59, 56, 56, 54, 47, 51, 64, 48, 59, 59, 53, 61, 55, 57, 63, 55, 56, 62, 57, 55)

puma_labor

##  [1] 56 51 62 58 59 59 56 56 54 47 51 64 48 59 59 53 61 55 57 63 55 56 62 57 55

(5 points) What’s the average, rounded to the nearest integer, that it took these 25 female pumas to give birth? Show your work.

round(mean(puma_labor))

## [1] 57

(10 points) How many standard deviations from the sample average is the most extreme value in the sample? Show your work.

sd(puma_labor)

## [1] 4.360046

(35 points) Test whether the mean labor time attained with this new program is less than the historical mean. Report the following as part of your answer: The hypotheses (Ho and Ha).

Ho: average (mean) labor >= 60 minutes

Ha: average (mean) labor < 60 minutes

The P value.

t.test(puma_labor, alternative= c("less"), mu=60)

## 
##  One Sample t-test
## 
## data:  puma_labor
## t = -3.9908, df = 24, p-value = 0.0002697
## alternative hypothesis: true mean is less than 60
## 95 percent confidence interval:
##     -Inf 58.0119
## sample estimates:
## mean of x 
##     56.52

Conclusion: p-value = 0.0002697 < alpha (0.05); We reject Ho (and support Ha)

(10 points) (This is the challenge question)

Note: Challenge questions are those I want you to answer without getting any help from me (i.e., you cannot ask me how to answer them). It is only 10 points; thus, do not worry. Try your best to answer it.

What type of statistical error could you potentially be making after your decision in part c? Is the probability that you are making this type of error small or large? Justify the answers to both of these questions.

It is a Type 1 error because we reject Ho and Ho could be true. The probability shows an small error due to alpha is 0.05 and we get a value lower than 0.05.