Statistics

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STATISTICS SET -1 S OLVED

Statistics For Management

1. (a) ‘Statistics is the backbone of decision-making’. Comment. (b) ‘Statistics is as good as the user’. Comment. Answer: (a) Statistics is the backbone of decision-making Due to advanced communication network, rapid changes in consumer behaviour, varied expectations of variety of consumers and new market openings, modern managers have a difficult task of making quick and appropriate decisions. Therefore, there is a need for them to depend more upon quantitative techniques like mathematical models, statistics, operations research and econometrics. Decision making is a key part of our day-to-day life. Even when we wish to purchase a television, we like to know the price, quality, durability, and maintainability of various brands and models before buying one. As you can see, in this scenario we are collecting data and making an optimum decision. In other words, we are using Statistics. Again, suppose a company wishes to introduce a new product, it has to collect data on market potential, consumer likings, availability of raw materials, feasibility of producing the product. Hence, data collection is the back-bone of any decision making process. Many organisations find themselves data-rich but poor in drawing information from it. Therefore, it is important to develop the ability to extract meaningful information from raw data to make better decisions. Statistics play an important role in this aspect. Statistics is broadly divided into two main categories. The two categories of Statistics are descriptive statistics and inferential statistics. • Descriptive Statistics :Descriptive statistics is used to present the general description of data which is summarised quantitatively. This is mostly useful in clinical research, when communicating the results of experiments. • Inferential Statistics :Inferential statistics is used to make valid inferences from the data which are helpful in effective decision making for managers or professionals. Statistical methods such as estimation, prediction and hypothesis testing belong to inferential statistics. The researchers make deductions or conclusions from the collected data samples regarding the characteristics of large population from which the samples are taken. So, we can say ‘Statistics is the backbone of decision-making’. (b) Statistics is as good as the user: Statistics is used for various purposes. It is used to simplify mass data and to make comparisons easier. It is also used to bring out trends and tendencies in the data as well as the hidden relations between variables. All this helps to make decision making much easier. Let us look at each function of Statistics in detail.

Statistics For Management

1. Statistics simplifies mass data The use of statistical concepts helps in simplification of complex data. Using statistical concepts, the managers can make decisions more easily. The statistical methods help in reducing the complexity of the data and consequently in the understanding of any huge mass of data. 2. Statistics makes comparison easier Without using statistical methods and concepts, collection of data and comparison cannot be done easily. Statistics helps us to compare data collected from different sources. Grand totals, measures of central tendency, measures of dispersion, graphs and diagrams, coefficient of correlation all provide ample scopes for comparison. 3. Statistics brings out trends and tendencies in the data After data is collected, it is easy to analyse the trend and tendencies in the data by using the various concepts of Statistics. 4. Statistics brings out the hidden relations between variables Statistical analysis helps in drawing inferences on data. Statistical analysis brings out the hidden relations between variables. 5. Decision making power becomes easier With the proper application of Statistics and statistical software packages on the collected data, managers can take effective decisions, which can increase the profits in a business. Seeing all these functionality we can say ‘Statistics is as good as the user’.

Statistics For Management

2. Distinguish between the following with example. (a) Inclusive and Exclusive limits. (b) Continuous and discrete data. (c) Qualitative and Quantitative data (d) Class limits and class intervals. Answer : a) Inclusive and Exclusive limits. Inclusive and exclusive limits are relevant from data tabulation and class intervals point of view. Inclusive series is the one which doesn't consider the upper limit, for example, 00-10 10-20 20-30 30-40 40-50 In the first one (00-10), we will consider numbers from 00 to 9.99 only. And 10 will be considered in 10-20. So this is known as inclusive series. Exclusive series is the one which has both the limits included, for example, 00-09 10-19 20-29 30-39 40-49 Here, both 00 and 09 will come under the first one (00-09). And 10 will come under the next one. b) Continuous and discrete data. All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get 0, 1, 2, 3, etc. All data that are the result of measuring are quantitative continuous data assuming that we can measure accurately. Measuring angles in radians might result in the numbers p/6, p/3, p/2, p/, 3p/4, etc. If you and your friends carry backpacks with books in them to school, the numbers of books in the backpacks are discrete data and the weights of the backpacks are continuous data. c) Qualitative and Quantitative data Data may come from a population or from a sample. Small letters like x or y generally are used to represent data values. Most data can be put into the following categories: • • Qualitative Quantitative

Statistics For Management

Qualitative data Qualitative data are the result of categorizing or describing attributes of a population. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O-, or B+. Qualitative data are not as widely used as quantitative data because many numerical techniques do not apply to the qualitative data. For example, it does not make sense to find an average hair color or blood type. Quantitative data Quantitative data are always numbers and are usually the data of choice because there are many methods available for analyzing the data. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and the number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous. All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get 0, 1, 2, 3, etc. Example 2: Data Sample of Quantitative Continuous Data The data are the weights of the backpacks with the books in it. You sample the same five students. The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data because weights are measured. Example 3: Data Sample of Qualitative Data The data are the colors of backpacks. Again, you sample the same five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. The colors red, black, black, green, and gray are qualitative data. Inclusive and Exclusive Class Intervals Inclusive Class Interval : When the lower and the upper class limit is included, then it is an inclusive class interval . For example - 220 - 234, 235 - 249 ..... etc. are inclusive type of class intervals. Usually in the case of discrete variate, inclusive type of class intervals are used. Exclusive Class Interval : When the lower limit is included, but the upper limit is excluded, then it is an exclusive class interval . For example - 150 - 153, 153 - 156.....etc are exclusive type of class intervals. In the class interval 150 - 153, 150 is included but 153 is excluded.

Statistics For Management

Usually in the case of continuous variate, exclusive type of class intervals are used. Consider the frequency table shown below

Note: While analysing a frequency distribution, if there are inclusive type of class intervals they must be converted into exclusive type. This can be done by extending the class intervals from both the ends. Thus the class intervals 220 - 234, 235 - 249, ....... should be converted into exclusive type 219.5 - 234.5, 234.5 - 249.5.... etc. After the conversion the frequency table would look like this

Statistics For Management

3. In a management class of 100 students’ three languages are offered as an additional subject viz. Hindi, English and Kannada. There are 28 students taking Kannada, 26 taking Hindi and 16 taking English. There are 12 students taking both Kannada and English, 4 taking Hindi and English and 6 that are taking Hindi and Kannada. In addition, we know that 2 students are taking all the three languages. i) If a student is chosen randomly, what is the probability that he/she is not taking any of these three languages? ii) If a student is chosen randomly, what is the probability that he/ she is taking exactly one language?

Let students taking Kannada as language be S(K) = 28 Let students taking Hindi as language be S(H) = 28 Let students taking English as language be S(E) = 28 Let students taking Kannada and English be S (K E ) = 12 Let students taking Hindi and English be S ( H E ) = 4 Let students taking Hindi and Kannada be S ( H K ) = 6 Let students taking all the three subjects be S (K H E ) = 2 If a student is chosen randomly, probability that he/she is not taking any of these three languages is P ( not taking any language) = [ 1 – { S(K)+ S(H)+S(E)+ S (K E )+ S ( H E )+ S ( H K )+ S (K H E} / 100 ] = [1 – {28+26+26+6+4+12+2} / 100] = 1 – (94 / 100) = 1 – 0.94 = 0.06 If a student is chosen randomly, probability that he/ she is taking exactly one language is P (taking exactly one language) = [ { S(K)+ S(H)+S(E) } ] / 100 = [28+26+16] = 70 = 70 = {70/100} = 0.7

Statistics For Management

4. List down various measures of central tendency and explain the difference between them? Measures of Central Tendency Several different measures of central tendency are defined below. 1 Arithmetic Mean The arithmetic mean is the most common measure of central tendency. It simply the sum of the numbers divided by the number of numbers. The symbol m is used for the mean of a population. The symbol M is used for the mean of a sample. The formula for m is shown below:

Where X is the sum of all the numbers in the numbers in the sample and N is the number of numbers in the sample. As an example, the mean of the numbers 1 + 2 + 3 + 6 + 8 = 20/5 = 4 regardless of whether the numbers constitute the entire population or just a sample from the population. The table, Number of touchdown passes (Table 1: Number of touchdown passes), shows the number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season. The mean number of touchdown passes thrown is 20.4516 as shown below.

Number of touchdown passes

Although the arithmetic mean is not the only "mean" (there is also a geometric mean), it is by far the most commonly used. Therefore, if the term "mean" is used without specifying whether it is the arithmetic mean, the geometric mean, or some other mean, it is assumed to refer to the arithmetic mean. 2 Median The median is also a frequently used measure of central tendency. The median is the midpoint of a distribution: the same number of scores are above the median as below it. For the data in the table, Number of touchdown passes (Table 1: Number of touchdown passes), there are 31 scores. The 16th highest score (which equals 20) is the median because there are 15 scores below the 16th score and 15 scores above the 16th score. The median can also be thought of as

Statistics For Management

the 50th percentile3. Let's return to the made up example of the quiz on which you made a three discussed previously in the module Introduction to Central Tendency4 and shown in Table 2: Three possible datasets for the 5-point make-up quiz. Three possible datasets for the 5-point make-up quiz

For Dataset 1, the median is three, the same as your score. For Dataset 2, the median is 4. Therefore, your score is below the median. This means you are in the lower half of the class. Finally for Dataset 3, the median is 2. For this dataset, your score is above the median and therefore in the upper half of the distribution. Computation of the Median: When there is an odd number of numbers, the median is simply the middle number. For example, the median of 2, 4, and 7 is 4. When there is an even number of numbers, the median is the mean of the two middle numbers. Thus, the median of the numbers 2, 4, 7, 12 is 4+7/2 = 5:5. 3 mode The mode is the most frequently occuring value. For the data in the table, Number of touchdown passes (Table 1: Number of touchdown passes), the mode is 18 since more teams (4) had 18 touchdown passes than any other number of touchdown passes. With continuous data such as response time measured to many decimals, the frequency of each value is one since no two scores will be exactly the same (see discussion of continuous variables5). Therefore the mode of continuous data is normally computed from a grouped frequency distribution. The Grouped frequency distribution (Table 3: Grouped frequency distribution) table shows a grouped frequency distribution for the target response time data. Since the interval with the highest frequency is 600-700, the mode is the middle of that interval (650). Grouped frequency distribution

Statistics For Management

Proportions and Percentages When the focus is on the degree to which a population possesses a particular attribute, the measure of interest is a percentage or a proportion. • A proportion refers to the fraction of the total that possesses a certain attribute. For example, we might ask what proportion of women in our sample weigh less than 135 pounds. Since 3 women weigh less than 135 pounds, the proportion would be 3/5 or 0.60. A percentage is another way of expressing a proportion. A percentage is equal to the proportion times 100. In our example of the five women, the percent of the total who weigh less than 135 pounds would be 100 * (3/5) or 60 percent.



Notation Of the various measures, the mean and the proportion are most important. The notation used to describe these measures appears below: • • • • • • X: Refers to a population mean. x: R efers to a sample mean. P: The proportion of elements in the population that has a particular attribute. p: The proportion of elements in the samplethat has a particular attribute. Q: The proportion of elements in the population that does not have a specified attribute. Note that Q = 1 - P. q: The proportion of elements in the samplethat does not have a specified attribute. Note that q = 1 - p.

Statistics For Management

5. Define population and sampling unit for selecting a random sample in each of the following cases. a) Hundred voters from a constituency b) Twenty stocks of National Stock Exchange c) Fifty account holders of State Bank of India d) Twenty employees of Tata motors. Statistical survey or enquiries deal with studying various characteristics of unit belonging to a group. The group consisting of all the units is called Universe or Population Sample is a finite subset of a population. A sample is drawn from a population to estimate the characteristics of the population. Sampling is a tool which enables us to draw conclusions about the characteristics of the population. In sampling there are two types namely discrete and the other is the continuous. Discrete sampling is that the data given are of the finite and their calculations are made easy. Continuous sampling is one where the data are of infinite form. Its intervals are indicated by <, >, greater than but lesser than, lesser than and greater than.. The finite number of items in a sample is size. in practice samples greater than 30 are large samples and if less it is small samples. A measure associated with the entire population is called as population parameter. Or just an parameter. Given a population, suppose we consider all possible samples of a certain size N that can be drawn from the population. For each sample suppose we compute a statistic such as mean, standard deviation etc. These sample vary from sample to sample. We group these different statistics according to their frequencies which is called as frequency distribution to formed so called as sampling distribution., standard deviation of a sampling distribution is called its standard error. Suppose we draw all possible samples of a certain size N from a population and find the mean of X bar of each of these samples. Frequency distribution of these means is called as sampling distribution of means. If the population is infinite , then , , be the standard deviation and mean respectively then the standard deviation denoted by is given by = / sqrt of N

is used to calculate the standard normal variate for the population where its size is more than 30.

Statistics For Management

6. What is a confidence interval, and why it is useful? What is a confidence level? Under a given hypothesis H the sampling distribution of a statistic S is a normal distribution with

the mean and the standard deviation then Z = is the standard normal variate associated with S so that for the distribution of z the mean is zero and the standard deviation is 1. Accordingly for z the Z% confidence level is ( -z c , zc) this means that we can be Z% confident that if the hypothesis H is true than the value of z lie between –zc and zc. This is equivalent saying that when H is true there is (100 –Z ) %chance that the value of z lies outside the interval (-zc . zc)if we reject a true hypothesis H on the grounds that the value of z lies outside the interval (-z, zc) we would be making a type 1 error and the probability of making this error is (100-Z)% the level of significance.

Confidence level is very much useful as we can predict any assumptions can be made so that it will not lead us to the wrong way even if it doesn’t be so great. As explained the confidence level is between –zc to z and the peak is at 100% which is the best. In some cases we predict but do not consider it , and sometimes we will not predict but hypothesis need it so this is called as the TYPE 1 errors and TYPE 2 errors. According to the levels of the Z the confidence is assured.. in the above the field shaded portion is the critical region.

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