Business Statistics: Descriptive And Inferential Statistics, Probability And Hypothesis Testing

What are Business Statistics?

Business statistics can be defined as a branch of statics that is applied to help ease business activities. The statistics is helpful in ensuring quality assurance in business. In addition, statistics allows for efficient financial analysis, production tracking and planning efficiency as well as several areas of business production (Morien, 2007). the general statistics is dived in two broad categories that is the descriptive and the inferential. Descriptive is applicable in explaining numbers while the inferential is applied to infer relationships from the parameters of the population.

Inferential statistics are more useful in cases where the business managers have limited data. For instance, the sampling, probability and models are useful for predicting future trends. Areas such as marketing largely depend on the inferential statistics due to the inability to involve the entire population in the research (Wegner, 2010). So as to gauge the future business budget requirements, the finance department usually applies statistical models to try represent a future occurrence.

Being that several business data come in form of numbers, descriptive statistics is a useful tool when it comes to breaking down such data. Statistical variables such as the mean, median and mode can assist the management monitor the prevailing activities as well as make informed business judgements (Nick, 2007). The use of ratios is occasionally applied to assist give highlight of the big picture.

Task 1: Selecting the random sample and creating the sample data file

The data to be statistically analysed composed of information about 400 people from a survey done in a unique State in the USA within a specific year. This population is comprised of working individuals who were drawing wages as at the time of the survey.

Before commencing the data analysis 50 random individual ID’s will be selected and the sample used for the purpose of the analysis. The sampling method will be without replacement where individual ID’s will only be selected once.

Definition of the variables;

V1: Wage

V2: Occupation

V3: Age

V4: Sex

1. Generation of column and pie chart

From the per chart and the column chart drawn above its evident that the sampled population is equally composed of the Female and Male workers (Novak, 2011). The visual display gives an indication of gender balance among the workforce.

Task 2: Probability and Binomial distribution

In the above table the frequency of each of the occupation practised within the state is indicated. It’s evident that the majority of the citizens do prefer other professions falling out of the management, sales, clerical, service and professional category. The state has a huge shortage of professionals with the sample data indicating no single professional is practising within the area.

Descriptive vs Inferential Statistics

The descriptive statistics presented indicates the features of the occupational category. The median is indicated as 3 which is the clerical profession. On average it can be stated that an individual picked at random within the streets of the state will be a clerk or a service provider.

Analysing the probabilities its evident that 28% of the working population are concentrated on the other occupations not mentioned (Manikandan, 2011). Sales and service do attract a lower probability if less than 20% while those employed under the professional occupation are accounted to by a zero probability.

The bar chart above gives a visual display of the probability distribution of the occupations.

1. A probability distribution is defined as an equation that links the association of an individual outcome of a statistical experiment with the occurrence probability. It is a definition of all the possible values of a random variable and their probabilities. Using the table below. It is possible to calculate.

1. Probability of 2

This is indicated as 0.1 and can be directly observed from the table.

1. Probability if more than 2

This is the chance that the value of x is neither 3, 4, 5 or 6.

Using probabilistic calculations, it can be obtained as 0.72.

• Probability of at least 3.

This the chance that the occurrence will either be 3,4,5 or 6.

Using probability calculations, it can be derived as 0.72

1. Using the prior information available from the previous research and the calculated probabilities it is possible to obtain posterior probabilities and calculate:
2. Probability if exactly two.

Through the excel calculations this is obtained as 0.1.

1. Probability of less than 2

The posterior probability is obtained as 0.18.

• Probability if at least 6. This gives the chance that the value is either 6 or above. From the calculations it is obtained as 0.28.

Task 3: Estimation and hypothesis testing

This is the application of statistics to obtain the chance that a given probability’s truth value is true. This process is usually characterised by four steps;

• Formulation of null hypothesis; This is stated as , is the indication that the observed features are due to pure chance and have no statistical relevance. On the other hand, the alternative hypothesis indicated as , this is the chance that the observations indicate a real effect combined with a component of chance variation.
• Identification of test statistic, this is the value that is applied in testing the truth value of the null hypothesis.
• Computation of the p_value, this is the probability that a test statistic at least as significant as the one observed will be obtained should the null hypothesis be true. A smaller p_value indicates a stronger evidence against the null hypothesis.
• Comparison of the p_value to the identified significance value . The decision criteria is that should the value of p be less or equals to , then the observed statistical effect is said to be significant (Sotos, Vanhoof, Noortgate, & Onghena, 2009). This way the null hypothesis is ruled
• Drawing the histogram

Histogram can be defined as an efficient representation of a numerical data’s distribution. The construction of a histogram involves first classifying the data in to class range before counting the number of variables that falls under each class. In this case the excel data analysis function was applied in generating the histogram.

From the data distribution presented by the histogram most of the workers wages in dollars per hour falls between $ 10 and $20. In addition, the range $ 10 to $ 40 accounts for the wage earned by 88% of the working population.

So as to test the notion that the mean of the population wage if $ 25, we use the one sample t test.

The test is carried out at an ,

The hypothesis to be tested is

Using the excel data analysis tool the output of the test is obtained as presented in the table below.

From the data above the two-tail p_value is obtained as 0.6864 which is greater than 0.01. In this case then we fail to reject the null hypothesis and conclude that; at a 99% confidence level its can be said that the mean of the population wage is $ 25.

1. The test carried out is a two tailed test. A two tailed test is a statistical test under which the critical area of a distribution has two sides and verifies whether a sample is greater than or less than a certain value. In the case above the alternative hypothesis is accepted should the mean either be less than or greater than 25. This way we can prove that the test is two tailed.
2. The null hypothesis is statistically written as

the statement means the mean of the population if 25.

On the other hand, the alternate hypothesis written as

mean of the population is not 25. This indicates that the mean can either be less than or greater than 25.

1. A t test is defined as a statistical test of hypothesis in which the test statistic follows the t distribution. The test is applicable in testing the significance of data.

Calculation of the t test statistic.

Replacing these values by the information obtained from the given data we have

1. The critical value is obtained by looking at the Z tables for the values of the alpha at the calculated degrees of freedom.

This is

1. From the data above the two-tail t stat is obtained as 0.4060 which is less than 2.67995. In this case then we fail to reject the null hypothesis and conclude that; at a 99% confidence level its can be said that the mean of the population wage is $ 25.
2. The 99% confidence interval is

Confidence Interval
wage (V1)
Confidence Level (99.0%)	6.024758494

1. At 5% level of significance

In this case the hypothesis being tested are:

This is a one tail test.

Being that the one tail p_value is greater than 0.05, we fail to reject the null hypothesis and conclude that the population mean is 25.

1. Considering the critical value, its is observed that t Critical One tail is 1.67655 which is greater than the t Stat, we thereby fail to reject the null hypothesis.
2. Finding the 95% confidence interval

95% confidence interval
wage (V1)
Confidence Level (95.0%)	4.517694943

1. The confidence interval indicates the range through which the mean of the population will fall. In this case it will be .

References

Manikandan, S. (2011). Frequency distribution. Journal of Pharmacology & Pharmacotherapeutics, vol. 2, No. 1, p. 54-55.

Morien, D. (2007). Business Statistics. Thomson Learning Nelson.

Nick, T. G. (2007). Descriptive Statistics: Topics in Biostatistics. Methods in Molecular Biology. New York: Springer.

Novak, S. Y. (2011). Extreme value methods with applications to finance. London: CRC/ Chapman & Hall/Taylor & Francis.

Sotos, A. E., Vanhoof, S., Noortgate, W. V., & Onghena, P. (2009). How Confident Are Students in Their Misconceptions about Hypothesis Tests? Journal of Statistics Education. Vol.17, No. 2.

Wegner, T. (2010). Applied Business Statistics: Methods and Excel-Based Applications. Juta Academic.