Statistic Problem Solution

Suppose there are two types of food: Slow and Fast. Slow food is food that takes a lot of time and energy to prepare, and is possibly more “authentic.” Consider it a high-end luxury good. All other food we’ll call Fast. It’s a staple needed to live, is quick to acquire and cheap. The slow food costs $20 per unit, while the fast food costs $5 per unit. The price elasticity of demand for slow food is -2.8. The price elasticity of demand for fast food is -.20. The government is considering a tax on food. The tax on slow food is denoted ts, and the tax on fast food is denoted tf.
a. Comment on the significance of the different elasticities of demand?
b. What is the equation for the optimal (Ramsey) value of ts in terms of tf.
c. In a clearly written paragraph, comment on the relative size of ts compared to tf, and why we are seeing this result.
d. Suppose the government has selected tax levels ts and tf using the Ramsey rule. Furthermore, at those taxes, the market sells 3 million units of slow food and 300 million units of fast food., the government collects $1 billion in revenue from these taxes. What are the values of th and tp?

Answer:

The different price elasticities of the two goods help in explaining the behavior of the two goods in the market. That is the responses of the quantity demanded with change in prices. Slow food shows that 1% change in price results to decline in quantity demanded by 2.8 units while fast foods shows that 1% change in price will results to decline in quantity by 2 units.

Optimal value of fast food VF= $5 – t f
       Optimal value for slow food VS=$20 – t s

The t s in fast moving foods is low because of the low price elasticity of demand of -2.0 while the t f in slow food is high due to high price elasticity of demand of -2.8.


300 units of fast foods
3 million units of slow foods
Rev 1 billion, therefore TS= 3/303 *1Billion=9900990
T f=300/303 * 1Billion=990099010

Frequency Distribution

Frequency Distribution of
Ungrouped and Grouped Data

Frequency distribution of ungrouped and grouped data is discussed below with examples.

Frequency distribution of ungrouped data:

Given below are marks obtained by 20 students in Math out of 25.

21, 23, 19, 17, 12, 15, 15, 17, 17, 19, 23, 23, 21, 23, 25, 25, 21, 19, 19, 19 

 

Frequency distribution of grouped data:

The presentation of the above data can be expressed into groups. These groups are called

classes or the class interval.

Each class interval is bounded by two figures called the class limits.

Marks 0 - 10 10 - 20 20 - 30

Number of Students (Frequency) 0 11 9

Note: The lower value of a class interval is called lower limit and upper value of that class interval is called the upper limit. Thus, each class interval has lower and upper limits.

For Example:

In the class interval 10 - 20, 10 is the lower limit and 20 is the upper limit.

Exclusive form of data:

This above table is expressed in the exclusive form.

In this, the class intervals are 0 - 10, 10 - 20, 20 - 30. In this, we include lower limit but exclude upper limit.

So, 10 - 20 means values from 10 and more but less than 20.

20 - 30 would mean values from 20 and more but less than 30.

Data in the inclusive form:

Marks obtained by 20 students of class VIII in Math text are given below.

23, 0, 14, 10, 15, 3, 8, 16, 18, 20, 1, 3, 20, 23, 24, 15, 24, 22, 14, 13

Let us represent this data in the inclusive form.

Marks 0 - 10 11 - 20 21 - 30

Number of Students (Frequency) 6 9 5

Here, also we arrange the data into different groups called class intervals, i.e., 0 - 10, 11 - 20, 21 - 30.

0 to 10 means between 0 and 10 including 0 and 10.

Here, 0 is the lower limit and 10 is the upper limit. 11 to 20 means between 11 and 20 including 11 and 20.

Here, 11 is the lower limit and 20 is the upper limit.

When the data is expressed in the inclusive form, it is converted to exclusive form by subtracting 0.5 from lower limit and adding it to upper limit of each class interval.

11 - 20 is expressed in the inclusive form which can be changed and taken as 10.5 - 20.5 which is the exclusive form of the data.

Similarly, 21 - 30 can be taken as 20.5 - 30.5.

Statistics is a branch of mathematics that deals with the interpretation of data. Statisticians work in a wide variety of fields in both the private and the public sectors.

In real life statistics, we come across numerical data in the newspapers, magazines and television regarding different aspects like increase or decrease in population, profit made by a company in different years, weather report, etc. These numerical facts are also represented by graphs which are easy to understand.

The term statistics is derived from a Latin word status meaning condition. The branch of mathematics which deals with the collection, presentation, analysis and interpretation of the numerical data is called statistics.

Terms Related to Statistics

Data:

The collection of information in the form of numerical figures, regarding different aspects of life is called data. The data can be about population, birth, death, temperature of place during a week, marks scored in the class, runs scored in different matches, etc. We need to analyze this data.

The following table gives the data regarding the number of students opting for different activities.

Activities	Dance	Music	Art	Sports
No. of students	15	25	10	40

Raw data:

When some information is collected randomly and presented, it is called a raw data.

For Example:

Given below are the marks (out of 25) obtained by 20 students of class VII A in mathematics in a test.

18, 16, 12, 10, 5, 5, 4, 19, 20, 10, 12, 12, 15, 15, 15, 8, 8, 8, 8, 16

Observation:

Each entry collected as a numerical fact in the given data is called an observation.

Array:

The raw data when put in ascending or descending order of magnitude is called an array or arrayed data.

For Example:

The above data is arranged in ascending order and represented as:

4, 5, 5, 8, 8, 8, 8, 10, 10, 12, 12, 12, 15, 15, 15, 16, 16, 18, 19, 20

Range:

The difference between the highest and the lowest value of the observation is called the range of the data.

In the above data,

Highest marks obtained = 20

Lowest marks obtained = 4

Therefore, range = 20 - 4 = 16

Mean:

It is calculated by dividing the sum of all the observation by the total number of observations. If x, x1, x3, ……… xn are n observations then

Arithmetic mean = (x1 + x2 + xn, ……………. xn)/n = (∑xi)/n

[∑ is the Greek letter sigma and is used to denote summation]

For Example:

The heights of 10 girls were measured in cm and results are as follows:

142, 149, 136, 148, 129, 140, 148, 145, 150, 133

(i) What is the height of the tallest girl?

Solution:

The height of the tallest girl is 150 cm.

(ii) What is the height of the shortest girl?

Solution:

The height of the shortest girl is 129 cm

(iii) What is the range of the data?

Solution:

Range = 150 cm – 129 cm = 21 cm

(iv) Find the mean height.

Solution:

The mean height = (142 + 149 + 136 + 148 + 129 + 140 + 148 + 145 + 150 + 133)/10

= 1420/10

= 142 cm

(v) How many girls are there whose height is less than the mean height?

Solution:

There are 4 girls whose height is less than the mean height, i.e., the girl having heights 136 cm, 129 cm, 133 cm, 140 cm.

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