Unit 5 Business Statistics and Research Methods PYQ’s 17th Oct 2020 Shift 2

The correct answer is – Chi – square test

The goodness of fit:

• The goodness of fit test is a statistical hypothesis test to see how well sample data fit a distribution from a population with a normal distribution.
• Put differently, this test shows if your sample data represents the data you would expect to find in the actual population or if it is somehow skewed.
• Goodness-of-fit establishes the discrepancy between the observed values and those that would be expected of the model in a normal distribution case.
• There are multiple methods for determining goodness-of-fit. Some of the most popular methods used in statistics include the chi-square, the Kolmogorov-Smirnov test, the Anderson-Darling test, and the Shapiro-Wilk test.
• Goodness-of-fit tests are often used in business decision making. In order to calculate a chi-square goodness-of-fit, it is necessary to first state the null hypothesis and the alternative hypothesis, choose a significance level (such as α = 0.5) and determine the critical value.

Key Points

1. The most common goodness-of-fit test is the chi-square test, typically used for discrete distributions.

2. The chi-square test is used exclusively for data put into classes (bins), and it requires a sufficient sample size in order to produce accurate results.

3. Goodness-of-fit tests are commonly used to test for the normality of residuals or to determine whether two samples are gathered from identical distributions.

Thus, option 3 is the correct answer.

Additional Information

1. T-Test:

• The t-test tells you how significant the differences between groups are; In other words it lets you know if those differences (measured in means) could have happened by chance.

• The t-test is one of many tests used for the purpose of hypothesis testing in statistics.

2. F Test:

• An F-test is conducted by the researcher on the basis of the F statistic.
• The F statistic is defined as the ratio between the two independent chi-square variates that are divided by their respective degree of freedom.
• The F-test follows the Snedecor’s F- distribution. The F-test is used by a researcher in order to carry out the test for the equality of the two population variances.
• If a researcher wants to test whether or not two independent samples have been drawn from a normal population with the same variability, then he generally employs the F-test.
• The F-test is also used by the researcher to determine whether or not the two independent estimates of the population variances are homogeneous in nature.

3. Z Test:

• A Z-test is a type of hypothesis test—a way for you to figure out if results from a test are valid or repeatable.
• For example, if someone said they had found a new drug that cures cancer, you would want to be sure it was probably true.
• A hypothesis test will tell you if it’s probably true, or probably not true. A Z test is used when your data is approximately normally distributed (i.e. the data has the shape of a bell curve when you graph it).

The correct answer is – Convenience sampling

Convenience sampling

1. It is defined as a method adopted by researchers where they collect market research data from a conveniently available pool of respondents.
2. It is the most commonly used sampling technique as it’s incredibly prompt, uncomplicated, and economical.
3. In many cases, members are readily approachable to be a part of the sample.
4. Researchers use convenience sampling in situations where additional inputs are not necessary for the principal research. There are no criteria required to be a part of this sample. Thus, it becomes incredibly simplified to include elements in this sample.
5. All components of the population are eligible and dependent on the researcher’s proximity to get involved in the sample.
6. The researcher chooses members merely based on proximity and doesn’t consider whether they represent the entire population or not.
7. Using this technique, they can observe habits, opinions, and viewpoints in the easiest possible manner.

In the question above, the researcher is choosing class friends and neighbors as his sample audience as it is convenient to get a response from them owing to their proximity. Thus, it is an example of Convenience Sampling.

Therefore, option 1 is the correct answer.

Additional Information

1. Judgement Sampling:

• Judgmental sampling, also called purposive sampling or authoritative sampling, is a non-probability sampling technique in which the sample members are chosen only on the basis of the researcher’s knowledge and judgment.
• The sample audience is selected with the utmost scrutiny and not as per convenience.

2. Cluster Sampling:

• Cluster sampling is a probability sampling technique where researchers divide the population into multiple groups (clusters) for research.
• Researchers then select random groups with a simple random or systematic random sampling technique for data collection and data analysis.

3. Stratified Sampling:

• Stratified sampling is a type of sampling method in which the total population is divided into smaller groups or strata to complete the sampling process.
• The strata are formed based on some common characteristics in the population data.
• After dividing the population into strata, the researcher randomly selects the sample proportionally.

The correct answer is – Longitudinal studies

Key Points

Longitudinal studies:

• A longitudinal study is a research design that involves repeated observations of the same variables i.e. people over a short or long period of time.
• It is a type of observational study, although they can be structured as longitudinal randomized experiments.
• The huge disadvantage of any longitudinal study is the time, as it typically takes a substantial time to collect the data that is required.
• Example: The 1970 British Cohort Study, which has collected data on the lives of 17,000 Brits since their birth in 1970, is one of the most well-known longitudinal studies.

Therefore, Longitudinal studies would be most appropriate to use in order to measure the changed opinions of consumers repeatedly.

Additional Information

Focus group studies:

• The focus group involves a small number of demographically similar individuals.
• It is a form of qualitative research where a group of people is asked about their opinions, beliefs, perceptions, and attitudes towards a product, service, advertisement, idea, concept, or packaging.
• The reaction of this group of people to specific research-posed questions are studied.

Shadowing studies:

• Shadowing is a qualitative research technique that is conducted on a small scale where the researcher acts as an observer.
• There are three types of shadowing:
• Natural – where the researcher only observes the subject for a set period of time without any interference.
• Controlled – where the researcher designs a task and only observes it being carried out.
• Participatory – where the researcher performs the activity being observed to gain a first-hand perspective.

Experimental studies:

• Experimental studies are the ones where the researcher introduces an intervention and then studies the effects.
• It is commonly used in scientific research designs and is the primary approach used to study the causal (cause and effect) relationship and investigate the relationship between one variable and the other.
• It is mostly done in a controlled environment.

The correct answer is – 0.1804

In probability theory and statistics, the Poisson distribution, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

Key Points

Using the Poisson probability rule,

P(X = x) = (λ^x.e^-λ) / x!

where, x = 0,1,2,3,…

λ = mean number of occurrences in the interval

e = Euler’s constant

Here, λ = 2 customers/minute. Now,

P(3) = (e^-2 x 2³) / 3!

With e-2 = 0.1353, we get

P(3) = (0.1353 x 8) / 6

∴ P(3) = 0.1804

Thus, option 1 is the correct answer.

The correct answer is – Mode = 3 Median – 2 Mean

Mean: By ‘mean’ we refer to an Arithmetic mean. The arithmetic mean refers to the average of a data set of numbers. It can either be a simple average or a weighted average. To calculate a simple average, we add up all the numbers given in the data set and then divided by how many numbers there are.

Median: Median is the middle number of a given data set when it is arranged in either a descending order or in ascending order. If there is an odd amount of numbers, the median value is the number that is in the middle whereas if there is an even amount of numbers, the median is the simple average of the middle pair in the dataset. Median is much more effective than a mean because it eliminates the outliers.

Mode: The mode refers to the number that appears the most in a dataset. A set of numbers may have one mode, or more than one mode, or no mode at all.

Key Points

The relationship between mean, median, and mode can be defined in four different cases:

In the case of a moderately skewed distribution, i.e. in general, the difference between mean and mode is equal to three times the difference between the mean and median. Thus, the empirical relationship can be written as:

Mean – Mode = 3 (Mean – Median)

∴ Mean – Mode = 3Mean – 3Median

∴ Mean – 3Mean = – 3Median + Mode

∴ – 2Mean = – 3Median + Mode

∴ Mode = 3Median – 2Mean.

Thus, Option 2 is the correct answer.

Additional Information

1. In the case of a frequency distribution that has a symmetrical frequency curve, the empirical relation states that mean = median = mode.
2. In the case of a positively skewed frequency distribution curve, mean > median > mode.
3. In the case of negatively skewed frequency distribution mean < median < mode.

The correct answer is – a-iv, b-iii, c-ii, d-i

Key Points

Therefore, it is clear from the above explanation that option 2) is the correct answer.

The correct answer is – B and C only

Forecasting is a technique that uses historical data as inputs to make informed estimates that are predictive in determining the direction of future trends.

Key Points

Problems in forecasting:

I Cyclical variation:

• Cyclical variations are long-term movements that represent consistently recurring rises and declines in the activity.
• These can be due to change in the business cycle as every organization passes from all the phases of the business cycle some time or the other.
• Prosperity, recession, depression, and recovery are the four phases of the business cycle.
• It is useful in formulating policies and forecasting activities.

II. Specification error:

• Specification error means that at least one of the key features or assumptions of the models is incorrect.
• As a consequence of the specification error, estimation of the model may give incorrect or misleading results.
• While forecasting, specification error can lead to problems as inaccurate estimates will be predicted.

Therefore, Statements B and C only include problems in forecasting.

Additional Information

I. Forecasting frequency:

• The length of a forecast and its frequency is interrelated.
• It is estimated that forecasts into the future are less reliable.
• Forecasts need to be updated on a regular periodic basis depending upon the change in the demand for the product.

II. Stationarity in data serires:

• Stationarity in data series refers to a process that has a property that the mean, variance, and autocorrelation structure do not change over time.
• A stationaries series is usually easy to predict as it is simply predicted that its statistical properties will be the same in the future.

III. Consistency in data series:

• Consistency refers to the quality or fact of remaining the same at different times.
• If there is consistency in a series of data, then the forecasting techniques adopted will be the same as it was earlier following the same pattern.

The correct answer is – A and C only

Customer interviews are a common mechanism for gathering the Voice of Customer (VOC). Customer interviews are usually conducted one-to-one with individual customers or with a small number of people from the same business or family group.

Key Points

Problems with customer interview:

A non-random sample:

• A non-random sample is a sampling technique where the sample selection is based on some factors other than some random chance.
• Non-random sampling relies on the subjective judgement of the researcher.
• It is a cost and time-consuming method, it is easy to use when there is a small population to work with.

Response bias:

• Response bias, also known as survey bias, is the tendency of a person to untruthfully or misleadingly answer questions of a survey.
• The bias can be intentional or accidental, but with biased responses, there is inaccuracy in the data gathered so the survey data becomes less useful.
• This can become a problem with customer interviews, especially with self-reporting participant surveys.

Therefore, a non-random sample and response bias could be problems with consumer interviews.