STS2003 Business Statistics for Managers
Assessed Learning Outcomes:
- Analyze the probability of given business outcomes using fundamental probability concepts for discrete and continuous probability distributions.
- Compute the probability of given business outcomes by building one and two tailed confidence intervals for Z and T probability distributions.
- Draw statistical inferences from given business information, based on one and two tailed hypothesis tests.
- Conduct hypothesis tests for the means of two or independent or dependent samples
You have been assigned a task on behalf of the statistical department of your college. You must undertake two hypothesis tests using a two sample approach. One must be dependent and the other must be independent. Your sample size should be between 15 and 30 observations per sample.
Task 1: In groups of no more than four people conduct your samples. The samples can be surveys, observations, oral questions. Please see below a list of possible samples you may use. One sample test must be independent and one must be dependent. You must write your data down on a notebook.
Task 2: Transfer this data in Excel (or another software package) then use appropriate functions to calculate the mean, standard deviation and the sample size. You may also use other basic Excel functions to get other data. Save your Excel files as some of the grades will be based on the output of the spreadsheets.
Task 3: Use appropriate statistical techniques to solve the questions you are being asked.
Task 4: You will do a short individual written defence (with the teacher’s data).This will be in class time and under examination conditions.
1.Write a short introduction to explain your two samples. You should describe who the population is (student and teachers) and why you chose this sample. You may also explain the sampling technique (random) you used to obtain your data.
2.In an appropriate software package calculate basic statistical data such as mean, standard deviation, median, range.
3.The data from one, two and three standard deviations from the sample mean.
4.Use the data to find the following information from your samples.
a)The top 15% of the data you have found in your sample.
b)The bottom 20% of the data you have found in your sample.
c)The middle 80% of the data you have found in your sample.
5.A) Find the 90%, 95% and 99% confidence intervals of all your data.
6.Use the two samples to set up a 5 step hypothesis comparing the two samples. You should use the following steps
a)Define your hypothesis
b)Choose 0.01, 0.05 or 0.10 significance levels and obtain critical values. Give reasons why they chose the level.
c)Do the test statistic
d)Formulate a decision on the findings of b) and c)
e)Give the result or decision of the hypothesis test and interpret what this means.
Answer:
Introduction
The given population of interest comprises of students of HCT, Oman and also teachers who are working full time at HCT. The objective of this task is to compare the monthly spending of two groups and considering that these groups ought to be independent, hence teachers and students were selected as the two groups from which the requisite sample has been collected. From the respective population of interest for each of the two samples, the sample comprising of 15 individuals has been selected using the simple random sampling. Using this particular sampling, 15 students and 15 teacher
s name were obtained and these were then contacted to obtain the required information about the monthly spending. Also, for the selected sample of teachers, the average monthly spending last year has also been obtained.
Basic Statistical Data
For the sample data collected, using Excel as the statistical package of choice, the basic statistical data has been obtained for the two samples. The output obtained is indicated below.
|
Monthly Spending (Rial) | |
|
Student |
|
|
Mean |
722.77 |
|
Standard deviation |
342.36 |
|
Sample Size |
15.00 |
|
Median |
655.00 |
|
Range |
1332.69 |
|
Monthly Spending (Rial) | ||
|
Teacher |
This year |
Last year |
|
Mean |
1438.67 |
1317.33 |
|
Standard deviation |
372.95 |
423.36 |
|
Sample Size |
15.00 |
15.00 |
|
Median |
1433.33 |
1346.67 |
|
Range |
1200.00 |
1566.67 |
Deviation from Mean
Based on the basis statistical data obtained for the respective samples, the following can be derived using Excel as the enabling tool.
|
STUDENT MONTHLY SPENDING |
Lower (Rial) |
Upper (Rial) |
|
Mean +/- 1 Standard deviation |
380.41 |
1065.13 |
|
Mean +/- 2 Standard deviation |
38.05 |
1407.48 |
|
Mean +/- 3 Standard deviation |
-304.30 |
1749.84 |
|
Monthly Spending of this year (Rial) | ||
|
Teachers |
Lower |
Upper |
|
Mean +/- 1 Standard deviation |
1065.71 |
1811.62 |
|
Mean +/- 2 Standard deviation |
692.76 |
2184.57 |
|
Mean +/- 3 Standard deviation |
319.81 |
2557.52 |
|
Monthly Spending of last year (Rial) | ||
|
Teachers |
Lower |
Upper |
|
Mean +/- 1 Standard deviation |
893.98 |
1740.69 |
|
Mean +/- 2 Standard deviation |
470.62 |
2164.05 |
|
Mean +/- 3 Standard deviation |
2587.40 |
2587.40 |
Percentile Computation
The percentile function has been used in Excel to compute the respective values from the given data of the two samples.
- a) The top 15% of the data would be represented by the 85 percentile.
85 percentile (Students) = 945.38 Rial
85 percentile (Teachers this year) = 1,793.33 Rial
85 percentile (Teachers last year) = 1,663.33 Rial
- b) The bottom 20% of the data would be represented by the 20 percentile.
20 percentile (Students) = 492.69 Rial
20 percentile (Teachers this year) = 1,049.33 Rial
20 percentile (Teachers last year) = 960 Rial
- c) The middle 80% of the data would be found between the 10 percentile and 90 percentile for the data.
Middle 80% (Students) = 705.31 Rial
Middle 80% (Teachers this year) = 868 Rial
Middle 80% (Teachers last year) = 926.67 Rial
Confidence Interval Computation
The confidence interval for student monthly spending is highlighted below.
The confidence interval for teacher monthly spending is represented below.
The confidence intervals are different above owing to the difference in t values based on the underlying confidence level. This confidence level is an indication of the precision required and typically higher is the precision required with the estimation of mean, the wider would be the confidence interval. As a result, the width of the confidence interval tends to increase as the confidence level increases (Flick, 2015).
Hypothesis Testing
1) Independent two sample t test
Step 1: Null Hypothesis: µstudent = µteacher
Alternative Hypothesis: µstudent ≠ µteacher
Step 2: The significance level has been chosen as 5%. For the given hypothesis, a level of 1% is not required. The given test would be 2 tailed considering the alternative hypothesis where “≠” is present. The critical values are -2.05 and 2.05 (Eriksson & Kovalainen, 2015).
Step 3: The relevant output obtained from Excel is indicated below.
It is apparent that the computed test statistic is -5.48.
Step 4: The computed test statistic does not tend to lie within -2.05 and 2.05. As a result, the available evidence would lead to rejection of null hypothesis and acceptance of alternative hypothesis (Hillier, 2016).
Step 5: Hence, it can be concluded that the monthly spending of students and teachers at HCT tends to differ in a statistically significant manner.
2) Dependent two sample t test
Step 1: Null Hypothesis: µlastyear = µcurrentyear
Alternative Hypothesis: µlastyear ≠ µcurrentyear
Step 2: The significance level has been chosen as 5%. For the given hypothesis, a level of 1% is not required. The given test would be 2 tailed considering the alternative hypothesis where “≠” is present. The critical values are -2.14 and 2.14 (Hair, Wolfinbarger, Money, Samouel, & Page, 2015).
Step 3: Considering that teachers are the same, hence the spending in current year and last year would be dependent. The relevant output obtained from Excel is indicated below.
Step 4: The computed test statistic tend to lie within -2.14 and 2.14. As a result, the available evidence would not lead to rejection of null hypothesis and acceptance of alternative hypothesis (Hillier, 2016).
Step 5: Hence, it can be concluded that the monthly spending of teachers at HCT in the current year and previous year does not tend to differ in any significant manner.
References
Eriksson, P. & Kovalainen, A. (2015) Quantitative methods in business research. 3rd ed. London: Sage Publications.
Flick, U. (2015) Introducing research methodology: A beginner's guide to doing a research project. 4th ed. New York: Sage Publications
Hair, J. F., Wolfinbarger, M., Money, A. H., Samouel, P., & Page, M. J. (2015) Essentials of business research methods. 2nd ed. New York: Routledge.
Hillier, F. (2016) Introduction to Operations Research. 6th ed. New York: McGraw Hill Publications
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