Business Statistics and Research Methods MCQ Quiz in मल्याळम - Objective Question with Answer for Business Statistics and Research Methods - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

The correct Answer is 86.67

Important Points Sampling distribution of mean based on a sample of 13 units.

This means

 \(n = 13\) which is less than 30, so we can use T-distribution.

\(N = 49\)

 \(\sigma ^2_\text {Sample variance}= 5\)5 

 \(\sigma^2 _\text{Pop variance}= ?\)  ?

\(\sigma ^2_\text {Sample variance}= \frac {\sigma^2 _\text{Pop variance}} {\text {Sample Size}} (\frac {N-n}{N-1})\)

\(5= \frac {\sigma^2 _\text{Pop variance}} {\text {13}} (\frac {49-13}{49-1})\)

\(\frac{5 \times 13 \times 48}{36}= {\sigma^2 _\text{Pop variance}}\)

\( {\sigma^2 _\text{Pop variance}} = 86.67\) (Answer)

The correct answer is Axiomatic probability

Key Points

Therefore, the correct answer is Option 3.

Chi-square test:

• A chi-square test, also written as χ2 test, is a statistical hypothesis test that is valid to perform when the test statistic is chi-square distributed under the null hypothesis, specifically Pearson's chi-square test and variants thereof.
• Pearson's chi-square test is used to determine whether there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table.
• In the standard applications of this test, the observations are classified into mutually exclusive classes.
• If the null hypothesis (that in the population there is no difference between the classes) is true, the test statistic computed from the observations follows a χ2 frequency distribution. 
• The purpose of the test is to evaluate how likely the observed frequencies would be assuming the null hypothesis is true.
• Test statistics that follow a χ2 distribution occur when the observations are independent and normally distributed, which assumptions are often justified under the central limit theorem.
• There are also χ2 tests for testing the null hypothesis of independence of a pair of random variables based on observations of the pairs.
• Chi-square tests often refer to tests for which the distribution of the test statistic approaches the χ2 distribution asymptotically, meaning that the sampling distribution (if the null hypothesis is true) of the test statistic approximates a χ2 distribution more and more closely as sample sizes increase

where:

Oi = an observed count for bin i

Ei = an expected count for bin i, asserted by the null hypothesis.

• Chi-square goodness of fit test determines if a sample data matches a population. 

Important Points

Goodness of fit:

• The goodness of fit of a statistical model describes how well it fits a set of observations.
• Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question

Use of Chi-square test:

• The chi-squared distribution has many uses in statistics, including:
• Confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation.
• Independence of two criteria of classification of qualitative variables.
• Relationships between categorical variables (contingency tables).
• Sample variance study when the underlying distribution is normal.
• Tests of deviations of differences between expected and observed frequencies (one-way tables).
• The chi-square test (goodness of fit test).

In the case of classification of data, the class having its upper limit is treated as the lower limit of its next class is called the Exclusive class.

Exclusive Class:

1. When the lower limit is included, but the upper limit is excluded, then it is an exclusive class interval.
2. For example: 150 - 153, 153 - 156.....etc. are an exclusive type of class intervals.
3. In the class interval 150 - 153, 150 is included but 153 is excluded. 153 is treated as the lower limit of the next class.
4. Usually, in the case of the continuous variate, exclusive type of class intervals are used.

1. Open-Ended Class: An open-ended distribution means that one or more classes (or bins) are open-ended. In other words, it doesn’t have a boundary. For instance, a height of 57″ or less is an open-ended distribution.

2. Close-Ended Class: The opposite would be a closed-ended distribution i.e when both the lower and upper limits are closed or bounded. For instance, an IQ of 142-149 is a close-ended distribution. 

3. Inclusive Class: 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 types of class intervals. Usually, in the case of the discrete variate, inclusive type of class intervals are used.

Thus, option 3 is the correct answer.

The normal distribution is the most important probability distribution in statistics because it fits many natural phenomena. For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution. It is also known as the Gaussian distribution and the bell curve.

Common Properties for All Forms of the Normal Distribution

1. Despite the different shapes, all forms of normal distribution have the following characteristic properties.
2. They’re all symmetric.
3. The normal distribution is a continuous probability distribution.
4. The normal distribution cannot model skewed distributions.
5. The mean, median, and mode are all equal.
6. Half of the population is less than the mean and half is greater than the mean.
7. The Empirical Rule allows you to determine the proportion of values that fall within certain distances from the mean.
8. A normal distribution curve is unimodal (it has only one mode).
9. The normal distribution has two parameters, the mean and standard deviation.

Thus, option 3 is not the correct property of normal distribution.

Below are the explanation of the above statements:

1.Deviation:

•  Method of Mean Deviation does not give accurate results. 
• The reason is that the mean deviation gives the best results when deviations are taken from the median.
• But the median is not a satisfactory measure when the degree of variability in a series is very high.
• And if we compute mean deviation from mean that is also not desirable because the sum of the deviations from the mean (ignoring signs) is greater than the sum of the deviations from the median (ignoring signs).
• If the mean deviation is computed from a mode that is also not scientific because the value of mode cannot always be determined. Thus, statement A is correct.

2. Standard deviation is independent of the change of origin and but not of scale. Thus, statement B is incorrect.

3. In a symmetrical distribution, the mean deviation equals 4/5 of standard deviation. Thus, statement C is correct.

4 In symmetrical and bell-shaped distribution, quartile deviation is the distance from the median to the lower quartile or to the upper quartile. Thus, statement D is incorrect. 

Correlational Research in psychology:

• In psychological research, we often wish to determine the relationship between two variables for prediction purposes.
• For example, you may be interested in knowing whether “the amount of study time” is related to the “student’s academic achievement”. you simply find out the relationship between the two variables to determine whether they are associated, or covary or not.
• The strength and direction of the relationship between the two variables are represented by a number, known as the correlation coefficient.

The correlation r measures the strength of the linear relationship between two quantitative variables. Pearson r:

\(r = {1 \over {n-1}} \sum ({x_{i} - {\bar x} \over S_{x}}) ({y_{i} - {\bar y} \over S_{y}})\)

• r is always a number between -1 and 1. Hence 1.20 does not lie in this range.
• r > 0 indicates a positive association.
• r < 0 indicates a negative association.
• Values of r near 0 indicate a very weak linear relationship.
• The strength of the linear relationship increases as r moves away from 0 toward -1 or 1.
• The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship.

Since -0.98 is close to -1 as compared between 0.90 and 1 therefore, -0.98 coefficients of correlation indicate the strongest relationship between two sets of variables.

The correct answer is Laspeyre's price index.

Key Points

• Laspeyre's price index:
  • This index uses the quantities from the base year as weights.
  • It measures the change in the cost of purchasing the base year's basket of goods in the current year.
  • Laspeyre's index tends to overstate inflation because it does not account for changes in consumption patterns when prices change.

Additional Information

• Walsch index:
  • This index uses the geometric mean of the base and current year quantities as weights.
  • It is not as commonly used as the Laspeyre's or Paasche's indices.
• Dorbish Bowley index:
  • This index is an average of the Laspeyre's and Paasche's indices.
  • It attempts to balance the upward bias of the Laspeyre's index and the downward bias of the Paasche's index.
• Paasche's price index:
  • This index uses the quantities from the current year as weights.
  • It measures the change in the cost of purchasing the current year's basket of goods in the base year.
  • Paasche's index tends to understate inflation because it accounts for changes in consumption patterns when prices change.

The correct answer is 13.6.

Key Points

•  The expected value of a binomial distribution is np, where n is the number of trials and p is the probability of success. In this case, n = 16 and p = 0.85, so the expected value is 16 * 0.85 = 13.6.
• This means that we can expect, on average, 13.6 successes in 16 trials. However, it is important to remember that the binomial distribution is a discrete distribution, so it is possible to get other results. For example, we could get 12 successes, 14 successes, or even 16 successes.

The probability of getting a particular number of successes can be calculated using the binomial formula.

The correct answer is Increasing returns to scale

Key Points

 Cobb-Douglas production function:

• The Cobb-Douglas production function is based on Paul H. Douglas and C.W. Cobb's empirical analysis of the American manufacturing industry.
• It is a degree one linear homogeneous production function that considers two inputs, labour and capital, for the total output of the manufacturing industry.
• The Cobb-Douglas production function is expressed as Q = AKaLß
  where Q is output and L and С are inputs of labour and capital, respectively. A, a and β are positive parameters where = a > O, β > O.

Important Points The Cobb Douglas production function exhibits the three types of returns:

• If a+b>1, there are increasing returns to scale.
• If a+b=1, we get constant returns to scale.
• If a+b<1, we get decreasing returns to scale.