Unit 5 Business Statistics and Research Methods PYQ’s 7th Dec 2023 Shift 1

The correct answer is 15.87%.

Key Points

• The Price Earnings (P/E) ratio is a metric that measures the value of a company, dividing the latest closing price of a company's stock by the earnings-per-share (EPS) over the last 12 months.
• A standardized value or Z-score is a statistical measurement that shows a score's relationship to the mean in a group of scores. A Z-score of 1.0 indicates a value that is one standard deviation from the mean.
• In a standard normal distribution:
  • About 68% of the values fall within one standard deviation of the mean (accounting for the area between Z= -1 and Z= 1).
  • About 95% fall within two standard deviations (Z=-2 to Z=2).
  • About 99.7 fall within three standard deviations (Z=-3 to Z=3).
• A Z-score of 1.0 indicates that the P/E ratio of the firm falls exactly one standard deviation above the mean in the distribution and is higher than about 84.13% of the values (because 50 percent are below the mean and an additional 34.13% lie between the mean and one standard deviation above).
• Answer options review:
  • 1. The value 99.73% corresponds to a Z-score of 3.0, not applicable here.
  • 2. 68.27% represents the total percentage of values within one standard deviation around the mean, not only above.
  • 3. 15.87% does not directly match with any typical Z-score, but if we consider that 84.13% of the firms have P/E ratio lower than this firm (due to its Z score of 1), it can be understood that this firm's P/E ratio is in the highest 15.87%.
  • 4. 34.13% represents the percentage of data between the mean and 1 standard deviation above only, not considering the portion (50%) under the mean.

Conclusion: The firm's P/E ratio is in the highest 15.87% of firms in the industry - Answer 3).

The correct answer is under no circumstances.

Key Points

• Analysis of Variance (ANOVA) is a statistical method that segregates observed variance into different parts for the purpose of assessing specific processes.
• The F-value in an ANOVA test is the test statistic for the hypothesis that the means of all groups (treatments) are equal.
• The F-value is calculated as the ratio between the Model Mean Square (or Treatment variance, which is computed as the Sum of Squares of the Treatment divided by the degrees of freedom) and the Error Mean Square (within treatment variability or error variance, computed as the Sum of Squares of Error divided by its degrees of freedom).
• Both the variance due to treatments (the numerator of the F ratio) and the error variance (the denominator of the F ratio) are sums of squares that are each divided by their respective degrees of freedom, and since these are sums of squares, they are always non-negative.
• Therefore, the F-value, being the ratio of these non-negative quantities, can never be negative.

The correct answer is A - IV, B - III, C - II , D - I.

Key Points

• A - IV: The time dimension - Cross-sectional study
  • Cross-sectional studies involve observing a particular outcome or phenomenon at a single point in time. This matches with "The time dimension" as it refers to how research is framed in terms of temporal aspects - in this case, at a specific point in time rather than longitudinally.
• B - III: The purpose of the study - Descriptive study
  • Descriptive studies aim to accurately and systematically describe a population, situation, or phenomenon. The purpose of such studies is often to gather more information about a topic when not much is known, matching "The purpose of the study."
• C - II: The topical scope of the study - Statistical study
  • A statistical study implies a focus on quantifying information and often involves the collection and analysis of numerical data across a scope defined by the study. "The topical scope of the study" would thus refer to what the study is trying to cover or address, which in statistical studies, is typically a defined research question or hypothesis assessed through quantitative means.
• D - I: The degree to which the research question has been crystallized - Exploratory study
  • Exploratory studies are conducted when a problem is not clearly defined. It investigates potential questions that could be researched more thoroughly and is utilized when the researcher does not know much yet about the problem. This corresponds with "The degree to which the research question has been crystallized," as exploratory studies are used when the research question is not yet fully formed or clear.

The correct answer is 0.67

Key Points

• The probability that a selected person is either a male or a smoker can be found using the formula P(A or B) = P(A) + P(B) - P(A and B), where A and B are two events. Here, A represents the event of being a male, and B the event of being a smoker.
• The probability of a person in the population being a male, P(A), is given as 57% or 0.57, and the probability of a person being a smoker, P(B), is given as 22% or 0.22. The probability of a person being both a male and a smoker, P(A and B), is given as 12% or 0.12.
• Substituting these probabilities into the formula gives P(A or B) = 0.57 + 0.22 - 0.12 = 0.67.
• Therefore, the probability that a randomly selected person is either a male or a smoker is 0.67 or 67%.

Additional Information

• Probability is a fundamental concept in mathematics and statistics that quantifies the likelihood or chance of an event occurring. It is a measure of uncertainty associated with a particular outcome in a given situation.
• Probability provides a numerical measure of uncertainty or randomness associated with different outcomes of a particular event or experiment. It helps us understand how likely or unlikely an event is to occur.

The correct answer is A, D, E Only 

Key Points

• Pre-experimental designs are the simplest form of research design and do not involve any kind of control or comparison group.
• Pre-experimental designs like the ones mentioned lack crucial elements such as random assignment, control groups, or pre-intervention measurements, which are essential for establishing causality and ruling out alternative explanations for observed effects.

 Important Points

• One-group pre test-post test: This design involves measuring a single group of participants before and after they have been exposed to some intervention or treatment. The main limitation of this design is the lack of a control group. Without a comparison group, it's challenging to attribute any observed changes solely to the intervention, as there could be other factors influencing the results.
• Static group comparison: In this design, two separate groups are compared, but participants are not randomly assigned to groups. One group receives the treatment (experimental group), and the other does not (control group). The major limitation of this design is the lack of random assignment, which means that the groups may differ systematically before the treatment, making it difficult to determine whether any differences observed after the treatment are due to the treatment itself or pre-existing differences between the groups.
• After-only study: This design involves measuring the outcome of interest only after the treatment has been administered, without any pre-intervention measurements or control group. The key limitation of this design is the absence of baseline data or a control group, making it challenging to establish causality or determine whether any observed outcomes are truly attributable to the treatment.

Correct answer is Assigns a probability to each possible value of the random variable.

Key Points

• Discrete probability distribution = A discrete probability distribution is a probability distribution that depicts the occurrence of discrete outcomes. In a discrete probability distribution each outcome has a specific probability assigned to it. A discrete probability distribution assigns a probability to each possible value of the random variable. Discrete distribution doesn't just range between -1 and +1 as it depends on the specific set of outcomes. There are 3 types of discrete probability distributions that are Binominal distribution, Poisson distribution and multi nominal distribution.
• Binominal distribution = Binominal distribution is also known as Bernoulli distribution. It is a discrete probability distribution expressing the probability of one set of dichotomous alternative that us success or failure.
  • Assumptions
    • An experiment is performed under the same conditions for a fixed number of trials say "n".
    • In each trial there are only two possible outcomes of the experiment that are success and failure.
    • The probability of success is denoted as p which remains constant from trial to trial on replacement. The probability of failure is denoted as q and is equal to (1-p).
    • The trials are statistically independent.
• Poisson distribution = It is a discrete probability distribution which was developed by French mathematician Simeon Devis Poisson. It may be expected in cases where the chance of any individual event being a success is small such as number of accidents on road. Also known as law of improbable events. All Poisson distribution are skewed to the right. Since the value of success is very small, the value of failure is almost equal to 1.
• Multi nominal distribution = Multi nominal distribution arises when the possibility of outcome is more than two. The probabilities of various outcomes remains the same for each trial, and the trials are independent. The sum of all the probabilities for all the outcomes in a multi nominal distribution is equal to 1.

Correct answer is A, B, D, and E Only 

Key Points

• Five number summary  = Five numbers summary is a part of the descriptive statistics and consist of five values and all these values help us to describe the data. This statistics helps to provide an overview of the central tendency, spread and overall shape of the data distribution. The summary consists of the following five statistics.
  • Median = The median is the middle value in a data when the data is ordered either in ascending or descending order. It separates the first half with the last half. In case of even number in the data the median is the average of the two middle values. It is simple to compute and is not affected by the extreme values in the dataset. It is appropriate average in dealing with the qualitative data that is where ranks are given. The disadvantages of using median is that its value is not determined by each and every observation and before computing the average it is necessary to arrange the data.
  • First quartile = First quartile is the value below 25% of the data in the distribution. In other words it is the 25th percentile. It is an essential part of the summary of five numbers as it provides an insight into the spread of the lower portion of the data distribution. First quartile is the median of the first half of the data. It is denoted as Q1.
  • Third quartile = Third quartile denoted as Q3 is the value below 75% of the data in the distribution. It is 75th percentile. It is also an important part of the summary of five numbers as it provides an insight into the spread of the higher portion of the data distribution. Along with the first quartile it can be used for calculating the interquartile range. Third quartile is the median of the upper half of the data.
  • Smallest value = Smallest value is the minimum value in the dataset. It provides the distribution about the lower floor of the data distribution.
  • Largest value = Largest value is the highest or the maximum value in the data set. It provides information about the upper bound or ceiling of the data distribution. The smallest value and the largest value both helps in understanding the extreme values and helps in assessing the range and variability of the dataset.

Correct answer is E, A, D, C, B

Key Points

• Process of a well planned experiment
  • Select relevant variables for testing = Anything which is capable of assuming the different values is known as variables. The first step in the process of an experiment is to identify the variables which are required for testing. There are mainly two types of variables dependent and independent variables.
    • Dependent variables = It is also called the outcome or response variables. This variable changes as a result of changes in the independent variables.
    • Independent variables = It is also known as manipulated, experimental or treatment variable. They become the cause of change in another variable.
    • In addition to these there are intervening and extraneous variables also. Intervening variables are those variables through which one variable affects another. Extraneous variable are those variable which affect the dependent variable and it needs to be controlled while conducting an experiment.
  • Specify the treatment levels = The next step is specifying the treatment levels means how the independent variable should be manipulated in the experiment and also controlling the extraneous variable so that the experimenter can determine weather the dependent variable changes in relation to a variation in an independent variable.
  • Chose the environmental and extraneous factors = Environmental and extraneous factors are those factors which can affect the dependent variable. Therefore the environmental and extraneous factors or the variable must be controlled so that the experimenter can determine weather the dependent variable changes in relation to a variation in an independent variable.
  • Choosing an experimental design = In this step an experimenter will choose an experimental design suited to the hypothesis. There are many experimental design which he can choose like cross sectional study( data is collected only once in the research process), longitudinal study(data is collected at various points of time in the research study), experimental study( in this the independent variable is manipulated to see the changes in the dependent variable), Action research( to solve the problem immediately).
  • Assign subject to group and conduct pilot study = The next step in the experiment is to assign the subjects or split the participants into groups like control and experimental group and then conduct a pilot study. Pilot study means to conduct an experiment on a small scale before conducting the main research to test the various aspects of the research design.

Key Points

• Statement I : For a given sample size and level of significance (a), the critical value of student's t always exceeds that of z. This is because the critical value of student's t distribution is greater than the critical value of of the standard normal distribution (z) when the degrees of freedom are relatively small.
• Statement II : For a given level of significance (a), the critical value of student's t decreases as the sample size increases, approaching the critical value of the standard normal distribution (z) as n becomes very large.

Additional Information

• The level of significance is the fixed probability of wrong elimination of null hypothesis when it is true. It defines whether the null hypothesis assumed is to be accepted or rejected.
• Critical value is defined as a value that is compared to a test statistic in hypothesis testing to determine whether the null hypothesis is to be rejected or not. The critical value for a one tailed or two tailed test can be calculated using the confidence interval.
• A t- test is used when the population standard deviation is not known and the sample size is less than 30.
• A z-test is used when the the population standard deviation is known and the sample size is greater than or equal to 30.