The histograms below summarize the average travel time to work for random samples of 70
workers in the Los Angeles area and 61 workers in the New York City area.
(a) Briefly compare the distributions of travel times for these two regions.
(b) Suppose you wanted to make the point that typical travel time is longer in the New York City area than in the Los Angeles area. Which measure of center would you use? Justify
your answer.

Answer:

(a) The travel times of the Los Angeles workers have a higher variability than those of the New York City area and about 54 out the 70 Los Angeles workers take longer (over 20minutes) than the maximum travelling time for the New York City workers (18 minutes) when they travel to work

(B) (i) The percentage of workers that travel close to the maximum travel times given in the data of the two cities

(ii) Almost half of the New York City area workers spend between 40 and 100 minutes to work, while almost half (32) of the Los Angeles workers spend between 0 and 20 minutes to work every day

Step-by-step explanation:

The travel times for the 70 Los Angeles workers are;

Number of workers                    [tex]{}[/tex]         Travel time

32                    [tex]{}[/tex]                                      0 - 20 minutes

22                    [tex]{}[/tex]                                      20 - 40 minutes

10                   [tex]{}[/tex]                                        40 - 60 minutes

4                   [tex]{}[/tex]                                          60 - 80 minutes

2                   [tex]{}[/tex]                                          80 - 100 minutes

The travel times for the 61 New York City area workers are;

18                   [tex]{}[/tex]                                          0 - 20 minutes

14                   [tex]{}[/tex]                                          20 - 40 minutes

10                   [tex]{}[/tex]                                          40 - 60 minutes

12                   [tex]{}[/tex]                                          60 - 80 minutes

7                    [tex]{}[/tex]                                           80 - 100 minutes

Workers in Los Angeles averagely take short time to reach to workplace than in New  York.

Descriptive statistics like mode, IQR, skewness etc do care about how the data is more leaning on one side than other. That is why these types of measures are helpful here.

Given data includes:

For Los Angeles: 32, 22, 10, 4, 2 (skewed left)

For New York: (18, 14, 10, 12 ,7) (very slightly skewed left)

Part a.

Comparing the distributions of travel times for both of the regions:

Histogram of Los Angeles is more skewed to the left. (values like 32, 20 lies on the left, and 5, 2 on the right tail)

Histogram of New York, in comparison to the histogram of Los Angeles is not that skewed and lies approximately leveled but is slight skewed to left.

This hows that time taken by worker to travel in Los Angeles is averagely very  short and count of workers taking longer time falls dramatically.

On the other hand, that above thing doesn't happen for workers in New York.

Part b.

Central measure "Mode" can be used here. The reason of choosing mode is that it focuses on frequency and gives weights to outliers too. Mean, on the other hand takes average of all values thus might forget the data of skewness.

The 3rd moment of measure of skewness can be used to measure the degree of skewness in both the data.

Inter Quartile Range is also useful since it will tell quartile to quartile measures which can help not loose the data of skewness.

So, the important thing that should be included in the measure we use to  describe this data is to include the information of skewness of the graph.

Descriptive statistics like mean, standard deviation etc forget about skewness.

Descriptive statistics like mode, IQR, skewness etc do care about how the data is more leaning on one side than other. That is why these types of measures are helpful here.

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