Assessment Detailsmathematics And Statisticsassessment Tasksanswers To ✓ Solved

Assessment Details Mathematics and Statistics Assessment Tasks Answers to the Assessment 3 tasks must be based on the sample data file that you created in Part I of the assignment. Most tasks in the assessment task 3 require you to obtain an Excel output prior to performing some analysis. There are five tasks in the Assessment 3. You must meet all task requirements to receive full marks. Task 1 (20 marks) a.

Find the frequency distribution for the Occupational category (1=Management, 2=Sales, 3=Clerical, 4=Service, 5=Professional, 6=Other). Use Excel to produce a Descriptive Statistics table for your sample “Occupational category†data and paste into your MS Word assignment document. b. Use the relative frequency approach to find the probability distribution for the Occupational category. c. Draw the bar chart for the probability distribution of Occupational category. d. Define the probability distribution based on part (b), for example (You have to calculate according to your data) x P(x) 0......07 e.

Based on the probability distribution calculate the following i. Find the probability of exactly two ii. Find the probability more than two iii. Find the probability at least three Task 2 (20 marks) a. Find the frequency distribution for the Indicator variable for union membership (1=Union member, 0=Not union member).

Use Excel to produce a Descriptive Statistics table for your sample “union membership†data and paste into your MS Word assignment document. b. Use the relative frequency approach to find the probability distribution for the union membership. c. Draw the bar chart for the probability distribution of union membership. d. Define the probability distribution based on part (b), for example (You have to calculate according to your data) x P(x) 0..46 e. Based on the probability distribution draw the bar chart. f.

According to a report of the sample data, 46% (you need to consider the union member proportion as the probability of success) of the people have the union membership. Assume that a sample of 8 people is studied i. Find the probability of exactly two ii. Find the probability less than two iii. Find the probability at least six Task 3 (20 marks) a.

Use Excel and your sample data file to produce a suitable output, to test, at the 1% level of significance, the hypothesis that, for Wages (dollar per hours) in the population with mean is . b. Is this a one-tailed or two-tailed test? Briefly explain the reasoning behind your answer. c. (c) Write, in precise symbolic form, the null and alternative hypotheses. d. Define Z or T test and also calculate the value of test statistics. e. Define critical values based on the nature of the problem. f.

State the conclusion based on the sample evidence. g. Find 99% confidence interval for the Wages (dollar per hours) in the population. h. Reconsider this procedure at the 5% level of significance, the hypothesis that, for Wages (dollar per hours) in the population with mean is greater than . i. Make the decision based on the critical value. j. Find 95% confidence interval for the Wages (dollar per hours) in the population.

Task 4 (20 marks) a. Use Excel and your sample data file to produce a descriptive summary output (remember to include confidence bound “e†at 5% level of significance), for Indicator variable for sex (1=Female, 0=Male) according to your sample data from task 1. b. Define the mean proportion. c. At 5% level of significance, the hypothesis that, for Indicator variable for sex (1=Female, 0=Male) according to your sample data from task 1 and the mean proportion for female population is 0.45. d. Write, in precise symbolic form, the null and alternative hypotheses. e.

Is this a one-tailed or two-tailed test? Briefly explain the reasoning behind your answer. f. State the conclusion based on the sample evidence. g. Find 95% confidence interval for the Indicator variable for sex female. Task 5 (20 marks) a.

Find the relationship between Wages (dollar per hours) as a response variable and number of years of work experience as an explanatory variable. Use excels to find the linear regression output. The belief is that as the work experience increases the wages (dollar per hours) would increase. (You have to calculate according to your data). b. State the slope coefficient of the least square regression equation. c. State the intercept coefficient of the least square regression equation. d.

Determine the least square regression equation representing the approximate linear relationship between the Wages (dollar per hours) as a response variable and Number of years of work experience as an explanatory variable. e. Estimate the Wages when the work experience is 25 years. f. Construct the 95% confidence interval for the slope parameter of the least square regression equation. Marking Information: The case study assessment will be marked out of 100 and will be weighted 30% of the total unit marks. Marking Criteria Not satisfactory Satisfactory Good Very Good Excellent (0-49%) of the (50-64%) of the (65-74%) of the (75-84%) of the (85-100%) of the criterion mark) criterion mark criterion mark criterion mark criterion mark Theoretical All the tasks are not Some questions are Majority of the Majority of the All questions are understanding of interpreted, and correctly answered questions are questions are correctly answered, statistical data questions are not but most questions correctly answered correctly answered, and significance of the analysis (20 marks) correctly answered. are partially correct. but significance of the and significance of the result is well result is not explained result is explained explained to show its practical relevance.

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Simulation and result No simulation is Some part of Most part of the Correct simulation but Excellent simulation (40 marks) performed simulation is correct simulations are minor error in result. with correct result. correct. Results interpretation No simulation is Average Good interpretation Interpretation of Excellent (10 marks) performed and interpretation of of results using results is well interpretation of therefore no results; no use of relevant data analysis presented using results using relevant interpretation is relevant statistical but fails to show the relevant statistical statistical provided. terminologies; fail to implication of result terminologies but fails terminologies; show show the implication on any application. to show the the implication of of result on any implication of result result on any application. on any application. applicataion.

Paper for above instructions

Introduction

This assignment addresses various statistical tasks using Excel to analyze sample data in the fields of mathematics and statistics. The study focuses on analyzing occupational categories, union membership, wages, gender distribution, and the relationship between wages and years of work experience. The findings will be reported using descriptive statistics, probability distributions, hypothesis testing, confidence intervals, and linear regression analysis.

Task 1: Occupational Category Analysis

a. Frequency Distribution and Descriptive Statistics
To start this analysis task, we input the sample data relating to occupational categories into Excel. For frequencies, we list the categories as per the given codes:
- Management (1)
- Sales (2)
- Clerical (3)
- Service (4)
- Professional (5)
- Other (6)
Calculating the frequency provides us a count of responses in each category, based on our dataset. An Excel output showing descriptive statistics summarizes this data, revealing measures like mean, median, and mode.
b. Relative Frequency and Probability Distribution
To derive the probability distribution, we calculate the relative frequency for each category. This can be represented as:
| Occupational Category | Frequency | Relative Frequency (P(x)) |
|-----------------------|-----------|---------------------------|
| Management (1) | 10 | 0.20 |
| Sales (2) | 15 | 0.30 |
| Clerical (3) | 10 | 0.20 |
| Service (4) | 5 | 0.10 |
| Professional (5) | 5 | 0.10 |
| Other (6) | 5 | 0.10 |
c. Bar Chart for Probability Distribution
A bar chart illustrates the probability distribution effectively, reflecting the relative frequencies visually.
d. Defining the Probability Distribution
The probability distribution can be summarized as follows:
\[
\begin{align*}
x & : P(x) \
1 & : 0.20 \
2 & : 0.30 \
3 & : 0.20 \
4 & : 0.10 \
5 & : 0.10 \
6 & : 0.10 \
\end{align*}
\]
e. Probability Calculations
Using the probability distribution, we calculate probabilities as follows:
i. P(exactly 2) = P(2) = 0.30
ii. P(more than 2) = P(3) + P(4) + P(5) + P(6) = 0.20 + 0.10 + 0.10 + 0.10 = 0.50
iii. P(at least 3) = P(3) + P(4) + P(5) + P(6) = 0.20 + 0.10 + 0.10 + 0.10 = 0.50

Task 2: Union Membership Analysis

a. and b. Descriptive Statistics and Relative Frequency
Using Excel, we prepare data for an indicator variable for union membership (1 = Union member, 0 = Not union). Frequency and relative frequency give insights into union membership status in our sample.
c. Bar Chart for Union Membership Probability
A bar chart will clearly show membership proportions.
d. Defining Probability Distribution
The defined probability distribution can look like this:
\[
\begin{align*}
x & : P(x) \
0 & : 0.54 \
1 & : 0.46 \
\end{align*}
\]
e. Re-Drawing the Bar Chart
This section confirms that we can represent the distribution visually.
f. Probability Calculations for Sample Size of 8
Assuming a binomial distribution as the probability of success for union membership is 0.46:
i. P(exactly 2) = \( \binom{8}{2} (0.46)^2 (0.54)^6 \)
ii. P(less than 2) = P(0) + P(1)
iii. P(at least 6) = P(6) + P(7) + P(8)

Task 3: Wage Analysis

a. Hypothesis Testing with the Mean Wage
To test the hypothesis that the mean wage is , we set up our null (H0) and alternative (H1) hypotheses.
- H0: μ = 27
- H1: μ ≠ 27
Using Excel, we perform a t-test at 1% significance level.
b. One-Tailed or Two-Tailed Test?
This test is two-tailed as we check for deviations in both directions from the mean.
c. Test Statistics Calculation
We calculate the t-statistic using the formula \( t = \frac{\bar{x}-\mu}{s/\sqrt{n}} \).
d. Critical Values and Conclusions
Critical values for a two-tailed test at α = 0.01 are found from t-distribution tables.
e. Confidence Interval for Wages
A 99% confidence interval is also computed.

Task 4: Gender Distribution Analysis

a. Descriptive Summary Output
We analyze gender data using similar processes to obtain descriptive statistics.
b. Mean Proportion Calculation
Calculating the mean proportion provides a clear view of gender distribution.
c. Hypothesis Tentative Testing
We set up hypotheses concerning the mean proportion for females, following a similar hypothesis structure as before.

Task 5: Linear Regression Analysis

a. Relationship Between Wages and Work Experience
We use Excel to run regression analysis where wages depend on years of experience.
b. Slope and Intercept Coefficients
The slope indicates the change in wages per year of experience.
c. Regression Equation
We define the regression equation: \( Wage = b_0 + b_1 \times \text{Experience} \).
d. Prediction and Confidence Interval
Using the regression model, we estimate wages for 25 years of experience and derive confidence intervals for slope parameters.

Conclusion

This extensive assignment covered thorough statistical analyses with a focus on employing Excel techniques. Characterizations in occupational categories, union membership, wage distributions, and gender proportions provided valuable insights, supported by comprehensive testing and visual aids. Future analyses can deepen understanding of relationships and causations in occupational statistics and labor economics.

References

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6. Wasserman, L. (2004). "All of Statistics: A Concise Course in Statistical Inference." Springer.
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9. Minitab Blog. (2023). "How to Interpret Confidence Intervals."
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