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 is(are) 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

n = 12

Let x be the times for 12 highscool students.

a)

Mode:

The mode is the number that appears most frequently in a data set.

Here 17 and 18 occurs two times in a datas sert.

Mode = 17, 18

b)
Mean:

Σε Mean n

297 Mean 12

Mean = 24.75

Mean = 24.75 minutes

Mean is a measure of central tendency.

c)

First arrange the data in increasing order:

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

n = 12 which is even.

Median = [(n/2)th observation + (n/2)+1 th observation ] / 2

= [6th observation + 7th observation ] / 2

= [19 + 23] /2

= 42/2

= 21

Median = 21

Median is a measure of central tendency.

d)
IQR (Interquartile range):

IQR = Q3 - Q1

Where Q3 is third quartile and Q1 is first quartile.

When number of observations is even. Q1 is simply the median of lower half of dataset and Q3 is simply the median of upper half of data set.

Lower half of dataset = {16, 17, 17, 18, 18, 19,}

Upper half of dataset = {23, 28, 29, 30, 34, 48}

Q1 = Median of {16, 17, 17, 18, 18, 19,}

= (17 + 18)/2

= 35/2

= 17.5

Q1 = 17.5

Q3 = Median of {23, 28, 29, 30, 34, 48}

= (29+30)/2

= 59/2

= 29.5

Q3 = 29.5

IQR = Q3 - Q1

IQR = 29.5 - 17.5

IQR = 12

IQR (Interquartile range) = 12

e)

If a data point is below Q1 – 1.5 * IQR or above Q3 + 1.5* IQR, it is viewed as an outlier.

Q1 – 1.5 * IQR = 17.5 - (1.5 * 12) = 17.5 - 18 = -0.5

Q3 + 1.5* IQR = 29.5 + (1.5 * 12) = 29.5 + 18 = 47.5

There is no data point below -0.5

The data point above 47.5 is 48

Thus 48 is on outlier.

Yes there is one outlier. 48 is outlier.

f)

Box plot is based on Five number summary.

Minimum, Q1, Q2, Q3, Maximum

Minimum = 16

Q1 = 17.5

Q2 = Median = 21

Q3 = 29.5

Maximum = 48

Box plot:

Mic=16 Max -48 9.-17.5 Q2 =21 Q₂ = 29.5

g)
Null hypothesis:

Ho :"> 28

Alternative hypothesis:

H, :μ< 28

h)
Yes we can use here t test since population standard deviation is not known.