Question 1
Question 1
Researchers are concerned with the decline in the proportion of koalas in the wild who are juveniles, as this will impact the future adult koala population. Historically, the proportion of koalas in South East Queensland who are juveniles is 20%.
Use the information in the dataset koalas17.sav to answer the following questions:
(a) Using SPSS, calculate the proportion of koalas in the study who are juveniles.
(b) Without using SPSS, determine whether there is evidence to support the theory that the proportion of koalas who are juveniles is less than 20%.
Perform a hypothesis test to statistically justify your answer by completing the following:
i. State the appropriate hypotheses (define any symbols used).
ii. Check the conditions and assumptions for this test.
iii. Calculate the test statistic for this test.
iv. Calculate the Pvalue for this test.
v. Interpret the Pvalue and write a meaningful conclusion in the context of this situation.
Question 2
Use the information in the dataset koalas17.sav to answer the following questions. You should use SPSS to calculate the sample statistics you will need to do this question, but for parts (b) and (c) you are required to do all other calculations by hand, using a calculator. Regardless of your answer to part (a), complete all parts of this question.
(a) Check the appropriate conditions and assumptions needed to calculate either a confidence interval or hypothesis test in relation to the population mean height of the trees in which juvenile koalas are sighted in South East Queensland. Comment on what these checks indicate about the appropriateness of proceeding with the analysis. Include an appropriate graph to support your answer.
(b) Estimate the population mean height of the trees in which juvenile koalas are sighted in South East Queensland, using a 95% confidence interval (show all working).
(c) From historical data, it is known that the mean height of trees in which juvenile koalas have been sighted is 15 metres. Perform a hypothesis test to see if there is evidence to support a suspicion that the mean height of trees in which juvenile koalas are sighted in South East Queensland is more than this.
In performing this test:
i. State appropriate hypotheses (define any symbols used).
ii. Calculate the value of a suitable test statistic for this test.
iii. Calculate the Pvalue of this test.
iv. Write a meaningful conclusion at the 5% level of significance.
Question 3
Use the information in the dataset koalas17.sav to answer the following questions. You should use SPSS to calculate any sample statistics you will need to do this question, but for parts b(ii) and (c) you are required to do all other calculations by hand, using a calculator.
Koalas of two broad age groups were sighted in South East Queensland – adult and juvenile. As a researcher you are interested to see whether there is a difference in the mean height of trees in which these two groups of koalas are sighted. Regardless of your answer to part (a), complete all parts of this question.
(a) Check the appropriate conditions and assumptions needed to perform a hypothesis test comparing the population mean heights of trees in which the two groups of koalas, based on age, are sighted. Comment on what these checks indicate about the appropriateness of proceeding with the analysis. Include an appropriate graph to support your answer.
(b) Using an appropriate statistical test, determine if, on average, there is a difference in the height of trees in which the two groups of koalas, based on age, are sighted in South East Queensland. In performing the test, include:
i. State appropriate hypotheses, clearly defining all symbols.
ii. Calculate a suitable test statistic (you can use the results from part (a) in this calculation.
iii. Find the Pvalue of the test (and include the degrees of freedom).
iv. Interpret the Pvalue and write a meaningful conclusion in the context of the question.
v. Now use SPSS to check your results for this hypothesis test. Attach or copy and paste the relevant output from SPSS for this test to your assignment.
vi. Briefly comment on how the test statistic and Pvalue from SPSS output are similar to or differ from your hand calculations.
(c) Estimate, with 90% confidence, the population mean difference in height of trees in which the two groups of koalas, based on age, are sighted in South East Queensland. Ensure you explain the confidence interval in the context of the question.
Question 4
Use the information in the dataset koalas17.sav to answer the following questions.
Researchers have theorized that juvenile koalas, when feeding, move down the tree rather than up. In order to test this theory, the height from the ground at which each juvenile koala was located sleeping within a tree was initially measured when the koala was sighted (positionintree) and then again three hours later (laterpositionintree). It now needs to be determined whether, on average, the height of the juvenile koalas in the trees has decreased after three hours.
(a) Use a parametric test to answer this question by completing the following (parts i. to v. are to be completed without the aid of SPSS, although summary statistics, i.e. mean and standard deviation, required for the test can be found using SPSS):
i. State appropriate hypotheses (define any symbols used).
ii. State (but do not check) the assumptions for carrying out this test. Describe the assumptions in the context of this question.
iii. Calculate the value of a suitable test statistic for this test.
iv. Calculate the Pvalue of this test.
v. Interpret the Pvalue and describe the outcome of the test in the context of this question.
vi. Now use SPSS to carry out the analysis. Copy and paste the relevant SPSS output to your assignment. Do these results agree with those found in part iv? (Hint: comment on the pvalue).
(b) Describe an alternative statistical test that could be used to answer this question.
Include in your answer:
i. the name of the test,
ii. the conditions/assumptions required for this test (in the context of the question),
iii. a definition of the test statistic that would need to be calculated to perform this test,
iv. the relative advantages and/or disadvantages of this test compared with the test you conducted to answer part (a),
v. the circumstances under which you would use this test in preference to the one used in part (a).
vi. Now use SPSS to carry out the analysis. Copy and paste the relevant SPSS output into your assignment.
vii. State and interpret the Pvalue from the SPSS output and describe the outcome of the test in the context of this question.
Question 1
Question 1
Researchers are concerned with the decline in the proportion of koalas in the wild who are juveniles, as this will impact the future adult koala population. Historically, the proportion of koalas in South East Queensland who are juveniles is 20%. Use the information in the dataset koalas17.sav to answer the following questions:
 Using SPSS, calculate the proportion of koalas in the study who are juveniles.
Solution
Table 1: Frequency distribution of age
Frequency 
Percent 
Valid Percent 
Cumulative Percent 

Valid 
Adult 
313 
86.5 
86.5 
86.5 
Juvenile 
49 
13.5 
13.5 
100.0 

Total 
362 
100.0 
100.0 
From table 1 above, the proportion of koalas in the study who are juveniles is 13.5%
 Without using SPSS, determine whether there is evidence to support the theory that the proportion of koalas who are juveniles is less than 20%. Perform a hypothesis test to statistically justify your answer by completing the following:
 State the appropriate hypotheses (define any symbols used).
Solution
 Check the conditions and assumptions for this test.
Solution
Because of the central limit theorem (large sample size), the sampling distribution of p is normally distributed
 Calculate the test statistic for this test.
Solution
 Calculate the Pvalue for this test.
Solution
The pvalue associated with the computed z score value is 0.000997.
 Interpret the Pvalue and write a meaningful conclusion in the context of this situation.
Solution
The pvalue is less than 5% level of significance we therefore reject the null hypothesis and conclude that that the proportion of koalas who are juveniles is significantly less than 20%.
Question 2
Use the information in the dataset koalas17.sav to answer the following questions. You should use SPSS to calculate the sample statistics you will need to do this question, but for parts (b) and (c) you are required to do all other calculations by hand, using a calculator. Regardless of your answer to part (a), complete all parts of this question.
 (7 marks) Check the appropriate conditions and assumptions needed to calculate either a confidence interval or hypothesis test in relation to the population mean height of the trees in which juvenile koalas are sighted in South East Queensland. Comment on what these checks indicate about the appropriateness of proceeding with the analysis. Include an appropriate graph to support your answer.
Solution
Normality Test
One of the key assumptions is related to the normality of the data. So we checked whether the variable height of the trees in which juvenile koalas are sighted in South East Queensland is normally distributed. Results are in table 2 below;
Table 2: Tests of Normality
Age 
KolmogorovSmirnov^{a} 
ShapiroWilk 

Statistic 
df 
Sig. 
Statistic 
df 
Sig. 

heightoftree 
Adult 
.074 
313 
.000 
.960 
313 
.000 
Juvenile 
.101 
49 
.200^{*} 
.954 
49 
.052 

*. This is a lower bound of the true significance. 

a. Lilliefors Significance Correction 
Using either KolmogorovSmirnov test or ShapiroWilk test, results shows that the variable height of the trees in which juvenile koalas are sighted in South East Queensland is indeed normally distributed (pvalue > 0.05).
Test of homogeneity of variance
Using Levene’s test, we checked whether the variable height has equal variance. As can be seen in table 3 below, the assumption on homogeneity is met (pvalue > 0.05). The two populations have equal variance.
Table 3: Test of Homogeneity of Variances
heightoftree 

Levene Statistic 
df1 
df2 
Sig. 
.416 
1 
360 
.519 
Other conditions that we found to have been met include
 Each value in the sample was sampled independently from each other value
 (6 marks) Estimate the population mean height of the trees in which juvenile koalas are sighted in South East Queensland, using a 95% confidence interval (show all working).
Solution
First we obtain the sample statistics.
Table 4: Descriptive Statistics
Age 
N 
Minimum 
Maximum 
Mean 
Std. Deviation 

Adult 
heightoftree 
313 
5 
50 
17.43 
5.390 
Valid N (listwise) 
313 

Juvenile 
heightoftree 
49 
6 
23 
15.86 
4.578 
Valid N (listwise) 
49 
Confidence interval;
Lower limit: 15.86  1.28184 = 14.57816
Upper limit: 15.86 + 1.28184 = 17.14184
Thus the 95% confidence interval is between 14.57816 and 17.14184
 (9 marks) From historical data, it is known that the mean height of trees in which juvenile koalas have been sighted is 15 metres. Perform a hypothesis test to see if there is evidence to support a suspicion that the mean height of trees in which juvenile koalas are sighted in South East Queensland is more than this. In performing this test:
 State appropriate hypotheses (define any symbols used).
Solution
 Calculate the value of a suitable test statistic for this test.
Solution
 Calculate the Pvalue of this test.
Solution
The pvalue associated with the z score value of 1.314985 is 0.094272.
 Write a meaningful conclusion at the 5% level of significance.
Solution
Solution to Question 1
Since the pvalue is greater than the 5% level of significance, we fail to reject the null hypothesis and conclude that there is no enough statistically significant evidence to conclude that the mean height of trees in which juvenile koalas sighted in South East Queensland is more than 15 metres.
Question 3
Use the information in the dataset koalas17.sav to answer the following questions. You should use SPSS to calculate any sample statistics you will need to do this question, but for parts b(ii) and (c) you are required to do all other calculations by hand, using a calculator. Koalas of two broad age groups were sighted in South East Queensland – adult and juvenile. As a researcher you are interested to see whether there is a difference in the mean height of trees in which these two groups of koalas are sighted. Regardless of your answer to part (a), complete all parts of this question.
 (6 marks) Check the appropriate conditions and assumptions needed to perform a hypothesis test comparing the population mean heights of trees in which the two groups of koalas, based on age, are sighted. Comment on what these checks indicate about the appropriateness of proceeding with the analysis. Include an appropriate graph to support your answer.
Solution
 The very first assumption that needs to be made is in regard to the scale of measurement that is applied on the collected data should be continuous or may be ordinal scale. The variable height is a continuous variable hence this assumption was met.
 The second assumption is in relation to randomness of the sample, the idea is that the collected data should be a representative data that is selected randomly from a portion of the total population. This assumption was also met.
 The third assumption is that the data follows a normal distribution, bellshaped distribution curve.
Results are in table 3 below;
Table 2: Tests of Normality
Age 
KolmogorovSmirnov^{a} 
ShapiroWilk 

Statistic 
df 
Sig. 
Statistic 
df 
Sig. 

heightoftree 
Adult 
.074 
313 
.000 
.960 
313 
.000 
Juvenile 
.101 
49 
.200^{*} 
.954 
49 
.052 

*. This is a lower bound of the true significance. 

a. Lilliefors Significance Correction 
Using KolmogorovSmirnov test, results shows that the variable height of the trees in which either adult or juvenile koalas are sighted in South East Queensland is indeed normally distributed (pvalue > 0.05).
 The other assumption is in relation to a reasonably large sample size. As could be seen, the sample size was 362 which is reasonably large hence this assumption was met.
 The last assumption is of equal variance i.e. homogeneity of variance. The data need to have equal variances.
Using Levene’s test, we checked whether the variable height has equal variance. As can be seen in table 3 below, the assumption on homogeneity is met (pvalue > 0.05). The two populations have equal variance.
Table 3: Test of Homogeneity of Variances
heightoftree 

Levene Statistic 
df1 
df2 
Sig. 
.416 
1 
360 
.519 
 Using an appropriate statistical test, determine if, on average, there is a difference in the height of trees in which the two groups of koalas, based on age, are sighted in South East Queensland. In performing the test, include:
 State appropriate hypotheses, clearly defining all symbols.
Solution
 Calculate a suitable test statistic (you can use the results from part (a) in this calculation.
Solution
Table 5: Descriptive Statistics
Age 
N 
Minimum 
Maximum 
Mean 
Std. Deviation 

Adult 
heightoftree 
313 
5 
50 
17.43 
5.390 
Valid N (listwise) 
313 

Juvenile 
heightoftree 
49 
6 
23 
15.86 
4.578 
Valid N (listwise) 
49 
 Find the Pvalue of the test (and include the degrees of freedom).
Solution
 Interpret the Pvalue and write a meaningful conclusion in the context of the question.
Solution
The pvalue associated with the z score value of 2.1761 is 0.0148. This value is less than the 5% level of signicance. We therefore reject the null hypothesis and conclude that there is a significant difference in the height of trees in which the two groups of koalas, based on age.
 Now use SPSS to check your results for this hypothesis test. Attach or copy and paste the relevant output from SPSS for this test to your assignment.
Solution
Using SPSS we obtain the following;
Group Statistics 

Age 
N 
Mean 
Std. Deviation 
Std. Error Mean 

heightoftree 
Adult 
313 
17.43 
5.390 
.305 
Juvenile 
49 
15.86 
4.578 
.654 
Independent Samples Test 

Levene's Test for Equality of Variances 
ttest for Equality of Means 

F 
Sig. 
t 
df 
Sig. (2tailed) 
Mean Difference 
Std. Error Difference 
95% Confidence Interval of the Difference 

Lower 
Upper 

heightoftree 
Equal variances assumed 
.416 
.519 
1.933 
360 
.054 
1.571 
.813 
.027 
3.169 

Equal variances not assumed 
2.177 
70.583 
.033 
1.571 
.721 
.132 
3.010 
An independent samples ttest was done to compare the mean height of trees for the adults and for the juvenile. Results showed that the average height of trees for the adults (M = 17.43, SD = 5.39, N = 313) had significant difference with the height of trees for the juvenile (M = 15.86, SD = 4.578, N = 49), t (320) = 1.93, p < .05, twotailed.
 Briefly comment on how the test statistic and Pvalue from SPSS output are similar to or differ from your hand calculations.
Solution
The test statistics from the SPSS is the same from that obtained from the hand calculations.
 (6 marks) Estimate, with 90% confidence, the population mean difference in height of trees in which the two groups of koalas, based on age, are sighted in South East Queensland. Ensure you explain the confidence interval in the context of the question.
Solution
Lower limit: 1.57 – 1.186836 = 0.383164
Upper limit: 1.57 + 1.186836 = 2.756836
Therefore the 90% confidence interval for the population mean difference in height of trees in which the two groups of koalas, based on age is between 0.3832 and 2.7568.
Question 4
Use the information in the dataset koalas17.sav to answer the following questions. Researchers have theorized that juvenile koalas, when feeding, move down the tree rather than up. In order to test this theory, the height from the ground at which each juvenile koala was located sleeping within a tree was initially measured when the koala was sighted (positionintree) and then again three hours later (laterpositionintree). It now needs to be determined whether, on average, the height of the juvenile koalas in the trees has decreased after three hours.
 Use a parametric test to answer this question by completing the following (parts i. to v. are to be completed without the aid of SPSS, although summary statistics, i.e. mean and standard deviation, required for the test can be found using SPSS):
 State appropriate hypotheses (define any symbols used).
Solution
Where,
 State (but do not check) the assumptions for carrying out this test. Describe the assumptions in the context of this question.
Solution
 The very first assumption that needs to be made is in regard to the scale of measurement that is applied on the collected data should be continuous or may be ordinal scale. The variable height is a continuous variable hence this assumption was met.
 The second assumption is in relation to randomness of the sample, the idea is that the collected data should be a representative data that is selected randomly from a portion of the total population. This assumption was also met.
 The third assumption is that the data follows a normal distribution, bellshaped distribution curve.
 The other assumption is in relation to a reasonably large sample size. As could be seen, the sample size was 362 which is reasonably large hence this assumption was met.
 The last assumption is of equal variance i.e. homogeneity of variance. The data need to have equal variances.
 Calculate the value of a suitable test statistic for this test.
Solution
Descriptive statistics
Descriptive Statistics 

Age 
N 
Minimum 
Maximum 
Mean 
Std. Deviation 

Adult 
Position of koala in tree (m) 
313 
4 
29 
13.14 
4.926 
Position of koala in tree 3h after first sighting (m) 
313 
0 
21 
11.02 
4.263 

Valid N (listwise) 
313 

Juvenile 
Position of koala in tree (m) 
49 
4 
21 
12.28 
4.276 
Position of koala in tree 3h after first sighting (m) 
49 
1 
19 
11.02 
4.323 

Valid N (listwise) 
49 
 Calculate the Pvalue of this test.
Solution
The pvalue related to the test statistics is 0.001759.
 Interpret the Pvalue and describe the outcome of the test in the context of this question.
Solution
The given pvalue is greater than the 5% level of significance we therefore reject the null hypothesis and conclude that on average, the height of the juvenile koalas in the trees has decreased after three hours (John , 2006).
 Now use SPSS to carry out the analysis. Copy and paste the relevant SPSS output to your assignment. Do these results agree with those found in part iv? (Hint: comment on the pvalue).
Solution
Using SPSS we obtained the following results;
Paired Samples Statistics 

Mean 
N 
Std. Deviation 
Std. Error Mean 

Pair 1 
Position of koala in tree (m) 
12.28 
49 
4.276 
.611 
Position of koala in tree 3h after first sighting (m) 
11.02 
49 
4.323 
.618 
Paired Samples Correlations 

N 
Correlation 
Sig. 

Pair 1 
Position of koala in tree (m) & Position of koala in tree 3h after first sighting (m) 
49 
.779 
.000 
Paired Samples Test 

Paired Differences 
t 
df 
Sig. (2tailed) 

Mean 
Std. Deviation 
Std. Error Mean 
95% Confidence Interval of the Difference 

Lower 
Upper 

Pair 1 
Position of koala in tree (m)  Position of koala in tree 3h after first sighting (m) 
1.255 
2.862 
.409 
.433 
2.077 
3.070 
48 
.004 
Yes the above results agree with those obtained by hand.
 Describe an alternative statistical test that could be used to answer this question. Include in your answer:
 the name of the test,
Solution
Wilcoxon Signed Rank Test
 the conditions/assumptions required for this test (in the context of the question),
Solution
The only condition for this test is that the values need to be numerical. It however does not rely on the assumptions that parametric tests need to have.
 a definition of the test statistic that would need to be calculated to perform this test,
Solution
The test statistics is given as;
 the relative advantages and/or disadvantages of this test compared with the test you conducted to answer part (a),
Solution
The main benefit of using Wilcoxon Signed Rank Test is the fact that it does not depends on the parent distribution nor on the parent parameters (Kerby, 2017). The assumptions on the distribution shape in not necessary when using this test.
 the circumstances under which you would use this test in preference to the one used in part (a).
Solution
This test can be used or applied anytime that the population of a given data cannot be assumed to be distributed normally..
 Now use SPSS to carry out the analysis. Copy and paste the relevant SPSS output into your assignment.
Solution
Ranks 

N 
Mean Rank 
Sum of Ranks 

Position of koala in tree 3h after first sighting (m)  Position of koala in tree (m) 
Negative Ranks 
31^{a} 
24.71 
766.00 
Positive Ranks 
14^{b} 
19.21 
269.00 

Ties 
4^{c} 

Total 
49 

a. Position of koala in tree 3h after first sighting (m) < Position of koala in tree (m) 

b. Position of koala in tree 3h after first sighting (m) > Position of koala in tree (m) 

c. Position of koala in tree 3h after first sighting (m) = Position of koala in tree (m) 
Test Statistics^{a} 

Position of koala in tree 3h after first sighting (m)  Position of koala in tree (m) 

Z 
2.821^{b} 
Asymp. Sig. (2tailed) 
.005 
a. Wilcoxon Signed Ranks Test 

b. Based on positive ranks. 
 State and interpret the Pvalue from the SPSS output and describe the outcome of the test in the context of this question.
Solution
The pvalue is 0.005, this value is less than 5% level of significance hence the null hypothesis is rejected and by rejecting the null hypothesis we conclude that the median height of the juvenile koalas in the trees has decreased after three hours.
References
John, R. A. (2006). Mathematical Statistics and Data Analysis. Journal of Statistical Computing, 4253.
Kerby, D. S. (2017). Comparing Two Samples from an Individual Likert Question. Journal of Statistical Theory and Practice, 5666.
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