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Q4(iii):
The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table:
(iii) Is it correct to conclude that the maximum number of leaves are 153 mm long? Why?
The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table:
(iii) Is it correct to conclude that the maximum number of leaves are 153 mm long? Why?
Solution :
Step 1: Data Pre-processing and Frequency Distribution Analysis
To analyze the distribution of leaf lengths, we must first examine the continuity of the class intervals provided in the data. The given data is as follows:
| Length (in mm) | Number of Leaves |
|---|---|
| 118 – 126 | 3 |
| 127 – 135 | 5 |
| 136 – 144 | 9 |
| 145 – 153 | 12 |
| 154 – 162 | 5 |
| 163 – 171 | 4 |
| 172 – 180 | 2 |
[Per the definition of grouped frequency distribution, intervals must be continuous to avoid data gaps. The gap between 126 and 127 is $127 - 126 = 1$ mm. The adjustment factor is $0.5$ mm.]
Step 2: Conversion to Continuous Intervals
To make the intervals continuous, subtract $0.5$ from the lower limit and add $0.5$ to the upper limit:
- $117.5 - 126.5$
- $126.5 - 135.5$
- $135.5 - 144.5$
- $144.5 - 153.5$
- $153.5 - 162.5$
- $162.5 - 171.5$
- $171.5 - 180.5$
Step 3: Evaluating the Claim
The question asks if it is correct to conclude that the maximum number of leaves are $153$ mm long. We analyze this based on the frequency table:
The modal class (the class with the highest frequency) is $144.5 - 153.5$, which contains $12$ leaves. This means there are $12$ leaves with lengths falling within the range of $144.5$ mm to $153.5$ mm.
[By the principles of Statistics, the frequency of a class interval represents the aggregate count of observations within that range, not the frequency of a specific discrete value.]
Step 4: Logical Deduction
The assertion that the "maximum number of leaves are 153 mm long" is scientifically and statistically incorrect for the following reasons:
- Range vs. Point: The data is grouped. We only know that $12$ leaves have lengths in the interval $[144.5, 153.5]$. We do not have information regarding the individual length of every leaf.
- Data Granularity: It is possible that none of the leaves are exactly $153$ mm long; they could be $145, 146, \dots, 152$ mm.
- Interpretation Error: The value $153$ is merely the upper boundary of the class interval with the highest frequency.
Final Solution: No, it is not correct to conclude that the maximum number of leaves are 153 mm long. The frequency of 12 corresponds to the entire interval of 144.5 mm to 153.5 mm, not to the specific value of 153 mm.
More Questions from Class 9 Mathematics Statistics EXERCISE 12.1
- Q1(i): A survey conducted by an organisation for the cause of illness and death among the women between the ages $15 - 44$ (in years) worldwide, found the following figures (in %) (i) Represent the information given above graphically.
- Q1(ii): A survey conducted by an organisation for the cause of illness and death among the women between the ages $15 - 44$ (in years) worldwide, found the following figures (in %): (ii) Which condition is the major cause of women’s ill health and death worldwide?
- Q1(iii): A survey conducted by an organisation for the cause of illness and death among the women between the ages $15 - 44$ (in years) worldwide, found the following figures (in %): (iii) Try to find out, with the help of your teacher, any two factors which play a major role in the cause in (ii) above being the major cause.
- Q2(i): The following data on the number of girls (to the nearest ten) per thousand boys in different sections of Indian society is given below. (i) Represent the information above by a bar graph.
- Q2(ii): The following data on the number of girls (to the nearest ten) per thousand boys in different sections of Indian society is given below. (ii) In the classroom discuss what conclusions can be arrived at from the graph.
- Q3(i): Given below are the seats won by different political parties in the polling outcome of a state assembly elections: (i) Draw a bar graph to represent the polling results.
- Q3(ii): Given below are the seats won by different political parties in the polling outcome of a state assembly elections: (ii) Which political party won the maximum number of seats?
- Q4(i): The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table: (i) Draw a histogram to represent the given data. [Hint: First make the class intervals continuous]
- Q4(ii): The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table: (ii) Is there any other suitable graphical representation for the same data?
- Q5(i): The following table gives the life times of 400 neon lamps: (i) Represent the given information with the help of a histogram.
- Q5(ii): The following table gives the life times of 400 neon lamps: (ii) How many lamps have a life time of more than 700 hours?
- Q6: The following table gives the distribution of students of two sections according to the marks obtained by them: Represent the marks of the students of both the sections on the same graph by two frequency polygons. From the two polygons compare the performance of the two sections.
- Q7: The runs scored by two teams A and B on the first 60 balls in a cricket match are given below: Represent the data of both the teams on the same graph by frequency polygons. [Hint : First make the class intervals continuous.]
- Q8: A random survey of the number of children of various age groups playing in a park was found as follows: Draw a histogram to represent the data above.
- Q9(i): 100 surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters in the English alphabet in the surnames was found as follows: (i) Draw a histogram to depict the given information.
- Q9(ii): 100 surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters in the English alphabet in the surnames was found as follows: (ii) Write the class interval in which the maximum number of surnames lie.
CBSE Solutions for Class 9 Mathematics Statistics
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