Question

The average number of times Americans dine out in a week fell from  in 2008 to  in 2012. The number of times a sample of  families dined out last week provides the following data. 7 2 6 4 8 4 1 4 2 4 5 2 3 5 2 0 6 7 4 2 a. Compute the mean and median. Mean (to 2 decimals) Median (to 1 decimal) b. Compute the first and third quartiles. Round the answers to 2 decimal places, if necessary. First quartile Third quartile c. Compute the range and interquartile range. Round the answers to 2 decimal places, if necessary. Range Interquartile range d. Compute the variance and standard deviation. Round the answers to 2 decimal places. Variance Standard deviation e. The skewness measure for these data is . Comment on the shape of this distribution. - Select your answer -The data are somewhat skewed to the rightThe data are somewhat skewed to the leftThe data's skewness is zeroItem 9 Is it the shape you would expect? - Select your answer -YesNoItem 10 Why or why not? The skewness measure - Select your answer -indicatesdoes not indicateItem 11 the direction of the data skew. f. Compute the lower limit and the upper limit (to 3 decimals and enter negative value as negative number.). Lower limit Upper limit Do the data contain outliers? - Select your answer -Yes, the data contain outliersNo, the data do not contain outliersItem 14

Answer 1

( a )

Mean :

Mean=Sum of terms / Number of terms

= 78 / 20

= 3.9

• Mean = 3.9

Median :

Ordering the data from least to greatest, we get:

0   1   2   2   2   2   2   3   4   4   4   4   4   5   5   6   6   7   7   8   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

Median= ( 4 + 4) / 2 = 4

• The median of the data set is 4.

( b )

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.

0   1   2   2   2   2   2   3   4   4   4   4   4   5   5   6   6   7   7   8   

So, the bottom half is

0   1   2   2   2   2   2   3   4   4   

• The first quartile of the data set is 2.

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.

0   1   2   2   2   2   2   3   4   4   4   4   4   5   5   6   6   7   7   8   

So, the upper half is

4   4   4   5   5   6   6   7   7   8   

• The third quartile of the data set is 5.5.

( c )

Range

The lowest value is 0.

The highest value is 8.

The range = 8 - 0 = 8.

• The range of the data set is 8

Interquartile range

The interquartile range is the difference between the third and first quartiles.

The third quartile is 5.5.

The first quartile is 2.

The interquartile range = 5.5 - 2 = 3.5.

• The interquartile range of the data set is 3.5.

( d )

Variance :

Create the following table.

data	data-mean	(data - mean)2
7	3.1	9.61
2	-1.9	3.61
6	2.1	4.41
4	0.1	0.01
8	4.1	16.81
4	0.1	0.01
1	-2.9	8.41
4	0.1	0.01
2	-1.9	3.61
4	0.1	0.01
5	1.1	1.21
2	-1.9	3.61
3	-0.9	0.81
5	1.1	1.21
2	-1.9	3.61
0	-3.9	15.21
6	2.1	4.41
7	3.1	9.61
4	0.1	0.01
2	-1.9	3.61

Find the sum of numbers in the last column to get.

∑(xi−X bar )²=89.8

Calculate variance using the above formula.

var= ∑(xi−X bar )² / n−1

=89.8 / ( 20−1 )

≈4.7263

• variance = 4.73

Standard deviation = sqrt ( 4.73 )

= 2.17

• Standard deviation = 2.17