Question

1. Consider the following times (rounded to the nearest minute) for 12 high school runners in a 5 K race 19 23 17 28 29 30 16 34 18 18 17 48 a. What isare) the mode(s)? b. What is the mean and what is it a measure of? c. What is the median and what is it a measure of? d. What is the value of the IQR? e. Are there any outliers according to the 1.5XIQR rule? Justify your answer. f. Make a boxplot of the data. g. If we wanted to determine whether the mean 5 K time all for high school runners is less than 28 minutes, what would our hypotheses be? (Don't forget to define the parameter of interest) h. Would you have any reservations about doing a t-test with the above data? Why or why not?

Answer 1

Note: Final answers are highlighted in colour.

a) Sorted list of numbers is

16,17,17,18,18,19,23,28,29,30,34,48.

Mode is the number that appears most of the times in the list.

17 and 18 have appeared 2 times each in the above list.

Therefore modes are 17 and 18

b) Mean is the measure of average time that runners take to finish the race.

Mean= Sum of all the numbers/ No of numbers=

=(16+17+17+18+18+19+23+28+29+30+34+48)/12

= 24.75

c)For even number of items in the data set, we compute the median by taking the mean of the two middlemost numbers in the sorted data set.

Sorted data set: 16,17,17,18,18,19,23,28,29,30,34,48.

Middle values are : 19, 23

Therefore , median= 19+23/2= 21

d)Summary of the data using R

> summary(mydata)
Min. 1st Qu. Median Mean 3rd Qu. Max.
16.00 17.75 21.00 24.75 29.25 48.00

From the above, Q1=17.75, Q3= 29.25

IQR= Q3- Q31= 29.25- 17.75=11.5

e) Outlier is any data point more than 1.5 interquartile ranges (IQRs) below the first quartile or above the third quartile.

Lower bound= Q1- I.5 *IQR= 17.75- 1.5*11.5= 0.5

Upper bound= Q3+I.5 *IQR= 29.25+1.5*11.5= 46.5

Outiers are numbers outside(0.5, 46.5)

Therefore 48 is an outlier.

Even for a naked eye, we can observe that 48 is very far from the other numbers.The next number smaller to 48 is 14 away from it.

f)

Boxplot of the data

boxplot(mydata,col = 'Blue',horizontal = TRUE)

15 20 25 30 35 40 45

g)The parameter of interest: The mean time to complete 5K.

Hypothesis:

H₀: $\mu_0 \geq 28$

H_a: $\mu_0 < 28$

h)The only reservation is, sample size of 12 is very less to perform a t test.

As we need to use central limit theorem to assume the normality of sampling distribution,sample size of 12 is insufficient and it gives a large standard error.