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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 P-value for this test.
v. Interpret the P-value 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 P-value 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 P-value of the test (and include the degrees of freedom).
iv. Interpret the P-value 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 P-value 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 P-value of this test.
v. Interpret the P-value 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 p-value).
(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 P-value 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:
  1. State the appropriate hypotheses (define any symbols used).

Solution

  1. 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

  1. Calculate the P-value for this test.

Solution

The p-value associated with the computed z score value is 0.000997.

  1. Interpret the P-value and write a meaningful conclusion in the context of this situation.

Solution

The p-value 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

Kolmogorov-Smirnova

Shapiro-Wilk

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 Kolmogorov-Smirnov test or Shapiro-Wilk test, results shows that the variable height of the trees in which juvenile koalas are sighted in South East Queensland is indeed normally distributed (p-value > 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 (p-value > 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:
  1. State appropriate hypotheses (define any symbols used).

Solution

  1. Calculate the value of a suitable test statistic for this test.

Solution

  • Calculate the P-value of this test.

Solution

The p-value associated with the z score value of 1.314985 is 0.094272.

  1. Write a meaningful conclusion at the 5% level of significance.

Solution

Solution to Question 1

Since the p-value 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, bell-shaped distribution curve.

Results are in table 3 below;

Table 2: Tests of Normality

Age

Kolmogorov-Smirnova

Shapiro-Wilk

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 Kolmogorov-Smirnov 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 (p-value > 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 (p-value > 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:
  1. State appropriate hypotheses, clearly defining all symbols.

Solution

  1. 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 P-value of the test (and include the degrees of freedom).

Solution

  1. Interpret the P-value and write a meaningful conclusion in the context of the question.

Solution

The p-value 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.

  1. 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

t-test for Equality of Means

F

Sig.

t

df

Sig. (2-tailed)

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 t-test 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, two-tailed.

  1. Briefly comment on how the test statistic and P-value 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):
  1. State appropriate hypotheses (define any symbols used).

Solution

Where,

  1. 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, bell-shaped 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

  1. Calculate the P-value of this test.

Solution

The p-value related to the test statistics is 0.001759.

  1. Interpret the P-value and describe the outcome of the test in the context of this question.

Solution

The given p-value 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).

  1. 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. (2-tailed)

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:
  1. the name of the test,

Solution

Wilcoxon Signed Rank Test

  1. 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;

  1. 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.

  1. 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..

  1. 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

31a

24.71

766.00

Positive Ranks

14b

19.21

269.00

Ties

4c

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 Statisticsa

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

Z

-2.821b

Asymp. Sig. (2-tailed)

.005

a. Wilcoxon Signed Ranks Test

b. Based on positive ranks.

  • State and interpret the P-value from the SPSS output and describe the outcome of the test in the context of this question.

Solution

The p-value 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, 42-53.

Kerby, D. S. (2017). Comparing Two Samples from an Individual Likert Question. Journal of Statistical Theory and Practice, 56-66.

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