Question

The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest. 16; 17; 19; 20; 20; 21; 22; 23; 24; 24; 24; 25; 25; 26; 26; 26; 27; 28; 29; 32; 33; 33; 34; 35; 37; 39; 40

a. Find the sample mean ?̅.

b. Find the mode.

c. Find the first quartile.

d. Find the median.

e. Find the third quartile.

f. Construct a box plot.

g. What percent of people surveyed visited a store at least 3 times?

h. Find the 40th percentile. i. Construct a histogram of the data.

j. Find the percentile of data point 29.

k. State whether the data are symmetrical, skewed to the left, or skewed to the right.

l. Find the sample standard deviation.

Answer 1

( a )  sample mean

The mean of a data set is the sum of the terms divided by the total number of terms. Using math notation we have:

Mean = Sum of terms / number of terms

= 725 / 27

= 26.8519

( b ) Mode :

The mode of a set of data is the value in the set that occurs most often.

Ordering the data from least to greatest, we get:

16   17   19   20   20   21   22   23   24   24   24   25   25   26   26   26   27   28   29   32   33   33   34   35   37   39   40   

Since both 24 and 26 occur 3 times, the modes are 24 and 26 ( this data set is bimodal).

( c ) First quartile :

The first quartile (or lower quartile or 25th percentile) is the median of the bottom half of the numbers. So, to find the first quartile, we need to place the numbers in value order and find the bottom half.

16   17   19   20   20   21   22   23   24   24   24   25   25   26   26   26   27   28   29   32   33   33   34   35   37   39   40   

So, the bottom half is

16   17   19   20   20   21   22   23   24   24   24   25   25   

The first quartile of these numbers is 22.

( d ) Median :

The median is the middle number in a sorted list of numbers. So, to find the median, we need to place the numbers in value order and find the middle number.

Ordering the data from least to greatest, we get:

16   17   19   20   20   21   22   23   24   24   24   25   25   26   26   26   27   28   29   32   33   33   34   35   37   39   40   

So, the median is 26 .

( e ) Third quartile

The third quartile (or upper quartile or 75th percentile) is the median of the upper half of the numbers. So, to find the third quartile, we need to place the numbers in value order and find the upper half.

16   17   19   20   20   21   22   23   24   24   24   25   25   26   26   26   27   28   29   32   33   33   34   35   37   39   40   

So, the upper half is

26   26   27   28   29   32   33   33   34   35   37   39   40   

The third quartile of these numbers is 33.