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PACC6008: Australian government sent out a survey in 2016 to a random sample of Australian workers - Statistics Assessment Answers

December 28, 2017
Author : Julia Miles

Solution Code: 1ABED

Question:

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Purpose: The purpose of this assignment requires you to understand the population & sample, identify the variable type, produce and interpret descriptive and inferential statistics and employ various graphical techniques to conduct quantitative research.

Question 1Suppose Australian government sent out a survey in 2016 to a random sample of Australian workers and collected various information of them, including gender, age, marital status, the number of hours watching TV each day, ethnicity, years of education, occupation, the number of hours spent on housework, the number of hours worked per week, work situation last week, income, satisfaction with life and travel time to work. The data is reported in the What_Analysis.xls data file.

(1) What is population Australian government interested in?

(2) Use Microsoft Excel to conduct and present the appropriate graphic and descriptive

analysis on two variables: income and ethnicity, interpret the results, and prepare a report.

(3) Suppose Australian government also conducted the same survey in 2015, the proportion of Australian workers who are satisfied with life was 0.65. They would like to know if this proportion has decreased.

  • What is the appropriate statistics used here to test it?
  • What are the null and alternative hypotheses for this test?
  • Produce the test results using Microsoft Excel, interpret it and make a final conclusion.

(4) Suppose Australian government also conducted the same survey in 2015, the average years of education Australian workers obtained was 10.8. They would like to know if this has changed.

  • What is the appropriate statistics used here to test it?
  • What are the null and alternative hypotheses for this test?
  • Produce the test results using Microsoft Excel, interpret it and make a final conclusion.

Question 2 Suppose you want to buy a two-bedroom apartment in Sydney and you are interested two main suburbs (Suburb 1 and Suburb 2) but you don’t know which one has higher average price. So you determined to do a research and see whether there is a significant difference of market price in these two suburbs. You are required to:

(1) Randomly select a sample of 50 properties in Suburb 1and another 50 properties in Suburb 2 and record their market prices. Thus, a total of 100 observations are recorded in your dataset. (2) What are populations you are interested in? (3) Conduct your analysis using hypothesis testing method.

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Solution:

Question 1

(1)

The population of interest in this study is the Australian workers. Here, the Australian government is interested in finding the factors that influence the life satisfaction of workers with life and travel time to work. For the purpose of this study, Australian workers are randomly selected and information such as Age, sex, marital status, hours spent in watching TV daily, race, educational qualification (in years), occupation, hours spent on housework, hours spent on job related works per week, status of work last week, income, life satisfaction and time taken to travel to work are recorded

(2)

The variable income is a continuous variable and hence descriptive statistics is computed for this variable and the ethnic is a nominal variable and therefore, frequency distribution is constructed for ethnic variable. The summary statistics and graphs for these two variables are given below

totinc
Mean 12760.6
Standard Error 641.5926
Median 8640
Mode 0
Standard Deviation 13503.96
Sample Variance 1.82E+08
Kurtosis 8.138731
Skewness 2.204028
Range 97550
Minimum 0
Maximum 97550
Sum 5652946
Count 443

Frequency Distribution – Ethnicity

Ethnic Frequency Percentage
Australian 286 64.71%
Northern Europe 96 21.72%
Southern Europe 38 8.60%
Other 22 4.98%
Total 442

Report

The mean Australian workers income (n = 443) is 12760.6 AUD with a standard deviation of 13503.96 AUD. The median income of Australian workers is 8640 AUD, indicating that, nearly 50% of sampled Australian workers income fall below 8640 AUD and 50% of sampled Australian workers income fall above 8640 AUD. The minimum and maximum recorded Australian worker income is 0 AUD and 97550 AUD respectively.

Regarding ethnicity, about 64.71% of the workers are Australians, 21.72% were from Northern Europe, 8.60% were from Southern Europe and the remaining 4.98 were from other ethnic background

(3)

In order to determine whether the proportion of Australian workers who are satisfied with life was decreased from 0.65, we perform one proportion z test

Null Hypothesis: H0: P >= 0.65

That is, the proportion of Australian workers who are satisfied with life was not decreased from 0.65

Alternate Hypothesis: Ha: P < 0.65

That is, the proportion of Australian workers who are satisfied with life was decreased from 0.65

Level of Significance

Let the level of significance be ? = 0.05

Test Statistic

The z test statistic is

Z=p-PP*(1-P)n

The table given below shows the workings of z test statistic

Data
Null Hypothesis p = 0.65
Level of Significance 0.05
Number of Items of Interest 263
Sample Size 436
Intermediate Calculations
Sample Proportion 0.603211009
Standard Error 0.0228
Z Test Statistic -2.0483
Lower-Tail Test
Lower Critical Value -1.6449
p-Value 0.0203
Reject the null hypothesis

From the above table, we see that the value of z test statistic is – 2.0483 and its corresponding p – value is 0.0203 < 0.05, indicating that there is sufficient evidence to reject the null hypothesis at 5% level of significance. Therefore, we conclude that the proportion of Australian workers who are satisfied with life was decreased from 0.65

(4)

Here, we wish to determine whether the average years of education Australian workers differ significantly from 10.8 and therefore, we use single mean z test to test the claim

Null Hypothesis: H0: µ = 10.8

That is, the average years of education Australian workers do not differ significantly from 10.8 years

Alternate Hypothesis: Ha: µ ? 10.8

That is, the average years of education Australian workers differ significantly from 10.8 years

Level of Significance

Let the level of significance be ? = 0.05

Decision Rule

  • If the p – value of t test statistic is less than 0.05, then there is sufficient evidence to reject the null hypothesis
  • If the p – value of t test statistic is greater than 0.05, then there is no sufficient evidence to reject the null hypothesis

Test Statistic

The z test statistic is

Z=x-?sn

The table given below shows the workings of z test statistic

Data
Null Hypothesis m= 10.8
Level of Significance 0.05
Population Standard Deviation 3.16
Sample Size 419
Sample Mean 11
Intermediate Calculations
Standard Error of the Mean 0.1544
Z Test Statistic 1.1409
Two-Tail Test
Lower Critical Value -1.9600
Upper Critical Value 1.9600
p-Value 0.2539
Do not reject the null hypothesis

From the above table, we see that the value of z test statistic is 1.1409 and its corresponding p – value is 0.2539 > 0.05, indicating that there is no sufficient evidence to reject the null hypothesis at 5% level of significance. Therefore, we conclude that the average years of education Australian workers do not differ significantly from 10.8 years

Question 2

(1)

The two suburbs taken into consideration are

  • Suburb1 (New Town, SA 5554)
  • Suburb 2 (HURSTVILLE, NSW 2220)

Convenience sampling technique was used to generate the random sample of 50 price information of properties from each suburbs. A convenience sample is a kind of non – probability sampling technique, which is easy to use and the researcher has the full freedom and convenient accessibility to select the subjects or respondents for his study.

(2)

The population of interest in this study is the property price or selling price of homes in Sydney.

(3)

In order to determine whether the mean price of two-bedroom apartment differs significantly in two suburbs in Sydney, we perform independent sample t test. The null and alternate hypotheses are given below

Null Hypothesis: H0: µ1 = µ2

That is, the mean price of two-bedroom apartment do not differ significantly in two suburbs in Sydney

Alternate Hypothesis: H0: µ1 ? µ2

That is, the mean price of two-bedroom apartment differ significantly in two suburbs in Sydney

Level of Significance

Let the level of significance be ? = 0.05

Decision Rule

  • If the p – value of t test statistic is less than 0.05, then there is sufficient evidence to reject the null hypothesis
  • If the p – value of t test statistic is greater than 0.05, then there is no sufficient evidence to reject the null hypothesis

Test Statistic

The t test statistic is

t=x1-x2s*1n1+1n2

The table given below shows the workings of independent sample t test statistic

t-Test: Two-Sample Assuming Equal Variances
Suburb1 (New Town, SA 5554) Suburb 2 (Hurstville, NSW 2220)
Mean 289651.9 1099296
Variance 22750776664 357329564473.47
Observations 50 50
Pooled Variance 190040170568.50
Hypothesized Mean Difference 0
df 98
t Stat -9.28627315
P(T<=t) one-tail 2.14332E-15
t Critical one-tail 1.660551217
P(T<=t) two-tail 4.28663E-15
t Critical two-tail 1.984467455

Going through the above table, we see that the value of t test statistic is -9.29 and its corresponding p – value is 0.000 < 0.001. Since the p – value of t test statistic is very low, there is sufficient statistical evidence to reject the null hypothesis at 5% level of significance. Therefore, we conclude that the mean price of two-bedroom apartment differ significantly in two suburbs in Sydney. In addition, the mean price of two bedroom apartment in suburb 1 is $ 289,651.9 and the mean price of two bedroom apartment in suburb 2 is $ 1,099,296. On comparing the mean prices, we can say that the mean price of two bedroom apartment in suburb 2 (HURSTVILLE, NSW 2220) is significant high when compared with the mean price of two bedroom apartment in Suburb1 (New Town, SA 5554). Therefore, the study findings suggests that the two bedroom apartment in Hurstville, NSW is five times costlier when compared to that of the price of two bedroom apartment in New Town, SA

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