Question

Consumer Reports posts reviews and ratings of a variety of products on its website. The following is a sample of 20 speaker systems and their ratings. The ratings are on a scale of 1 to 5, with 5 being best.

Speaker                                        Rating                     Speaker                             Rating

Infinity Kappa. 6.1                       4.00                ACI Sapphire III                         4.67

Allison One                                 4.12                Bose 501 Series                         2.14

Cambridge Ensemble II                3.82                DCM KY-212                             4.09

Dynaudio Contour 1.3                 4.00                Eosone RSF 1000                       4.17

Hsu Rsch. HRSW 12V                   4.56                Joseph Audio RM7 si                 4.88

Legacy Audio Focus                     4.32                Martin Logan Aerius                  4.26

Mission 73Ii                                4.33                Omni Audio SA 12.3                  2.32

PSB 400i                                     4.50                Polk Audio RT 12                       4.50

Snell Acoustics D IV                     4.64                Sunfire True Subwoofer             4.17

Thiel CS 1.5                                 4.20                Yamaha NS- A636                      2.17

1. Compute the mean and the median.
2. Compute the first and third quartiles.
3. Compute the standard deviation.
4. The skewness of this data is – 1.67. Comment on the shape of the distribution.
5. What are the z-scores associated with Allison One and Omni Audio?
6. Do the data contain any outliers? Explain.

Answer 1

Answer:

Consumer Reports posts reviews and ratings of a variety of products on its website. The following is a sample of 20 speaker systems and their ratings. The ratings are on a scale of 1 to 5, with 5 being best.

Speaker                                        Rating                     Speaker                             Rating

Infinity Kappa. 6.1                       4.00                ACI Sapphire III                         4.67

Allison One                                 4.12                Bose 501 Series                         2.14

Cambridge Ensemble II                3.82                DCM KY-212                             4.09

Dynaudio Contour 1.3                 4.00                Eosone RSF 1000                       4.17

Hsu Rsch. HRSW 12V                   4.56                Joseph Audio RM7 si                 4.88

Legacy Audio Focus                     4.32                Martin Logan Aerius                  4.26

Mission 73Ii                                4.33                Omni Audio SA 12.3                  2.32

PSB 400i                                     4.50                Polk Audio RT 12                       4.50

Snell Acoustics D IV                     4.64                Sunfire True Subwoofer             4.17

Thiel CS 1.5                                 4.20                Yamaha NS- A636                      2.17

(a)

Number of data points, n = 20

Mean=79.86/20=3.993
Arranging the data in ascending order provides the following list.
2.14,2.17,2.32,3.82,4.00 .4.09 ,4.12 ,4 17,4.17,4.2,4.26,4.32,4.33,4.5,4.5, 4.56, 4.64 ,4.67, 4.88.

Because n = 20 is even we identify the middle two values 4 17 and 4 2. The median is the
average of these values
Therefore Median =4.17 + 4.2/2=4.185

(b)

We know that 25th percentile is the first quartile

i=(p/100)n=(25/100)20=5

Because is an integer, the 25th percentile is the average of the 5th and 6th data values. Thus
the 25th percentile is

(4+4)/2=4
We know that 75th percentile is the third quartile

i=(p/100)n=(75/100)20=15

Because is an integer the 75th percentile is the average of the 15th and 16th data values. Thus
the 75th percentile is

(4.5+4.5)/2=4.5

(C)

The sum of squared deviations about mean is, =12.508

The sample variance is,

s²=/(n-1)=12.508/19

s²=0.6583

The sample standard deviation is,

=0.8114

(d)

Since skew ness is negative that there are more negative deviations than positive deviations
from mean It is highly skewed to the right Therefore the distribution is highly skewed to the night

(e)

Z-Score for Allison One is,
z- Score for Omni Audio is,
(f)

Z- Score for Joseph Audio is,

z - Score for Bose 501 series is

The z - scores of the largest and smallest values are within the three standard deviations. Hence
there is no outlier in the data.