401077 Introduction To Biostatistics

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30-07-2021

Questions:

Question 1

Research question: Does the average minutes of moderate to vigorous physical activity (MVPA) differ between overweight and non-overweight University students?

Use the assignment data set assigned to you: Variables to analyse: ‘MVPA’ and ‘overweight’

Note: Each student will get different answers as the data sets differ.

Draw histograms of MVPA by overweight status. Add reasonable axis labels.
Name the statistical test you would you use to address the above research question using the data set provided. Explain why this test is appropriate.
Answer the research question by conducting your chosen statistical test on the data set provided. Please use R Commander but format your answer following the 5 step method.

Question 2

Consider the research question “Is the mean weight of Australian university students heavier at graduation than at enrolment?” Suppose the following data was collected to address this hypothesis.

student ID	weight at enrolment (kgs)	weight at graduation (kgs)
1	86	89
2	64	73
3	51	55
4	73	72
5	63	80
6	69	70

Identify the most appropriate statistical test for addressing the research question using the given data. Explain why this test is appropriate. (Note: Please do not run any analyses or conduct the statistical test.)

Question 3

Research question: How different are the average GPAs of overweight and non-overweight Australian University students?

Use the assignment data set assigned to you: Variables to analyse: ‘GPA’ and ‘overweight’

Note: Each student will get different answers as the data sets differ.

Using R Commander, calculate the 95% confidence interval for the difference in mean GPA between the overweight and non-overweight groups.
Carefully write in words, what this confidence interval is telling us.
From the results in a) can we conclude that there a statistically significant difference in mean GPA between overweight and non-overweight Australian university students? Explain why or why not.

Question 4

Estimate the minimum sample size required to detect a difference of 0.5 in mean GPA between the overweight and non-overweight retirees with and power=0.80 . (That is, this difference could be between 4 and 4.5 or between 4.5 and 5.0. You can choose any means as long as they are 0.5 apart.) Assume the population standard deviation is and equal group sizes. Present your answer as a sentence which summarises the required sample size to achieve what power subject to what conditions.

Question 5

Research question: Does the proportion of overweight students differ by sex in the population of Australian university students?

Use the assignment data set assigned to you: Variables to analyse: ‘overweight’ and ‘sex’

Note: Each student will get different answers as the data sets differ.

Show the relationship between overweight and sex using a two way contingency table. Include either row or column percentages. Type and label the table yourself: an R Commander screenshot will not be accepted.
Looking at the results in part a) only, is there any evidence of association between sex and overweight status in this sampleof Australian university students? Explain why or why not.
Are the requirements for a Chi-square test met? Explain why.

Irrespective of your answer in part c) address the research question using a Chi-square test on the provided data. Please use R Commander but format your answer following the 5 step method.

Question 6

Estimate the minimum sample size required to produce a 95% confidence interval with a margin of error of for the proportion of Australia university students. Assume that 40% of this population will be overweight. Present your answer as a sentence which summarises the required sample size to achieve what confidence level subject to what conditions.

Answers:

Question 1

Figure 1: Histogram of MVPA of Overweight and non-overweight Australian university students

From the histograms we find that they are right skewed. Hence the mean MVPA of the overweight the respondents is less than median value. Similarly the mean MVPA of non-overweight respondents is less than median value.

To analyse the distribution of MVPA according to overweight the independent sample t-test is used. In the present analysis there are two unrelated groups – overweight and non-overweight. The overweight classification is the independent variables. We analyse the difference in overweights dependence on MVPA. MVPA is dependent variable.

Null Hypothesis The mean MVPA of overweight respondents is equal to the mean MVPA of non-overweight respondents

Alternate Hypothesis The mean MVPA of overweight respondents is not equal to the mean MVPA of non-overweight respondents

The test is carried out at a = 0.05

Thus if p-value for the test is less than a then accept Null Hypothesis else reject Null Hypothesis.

For overweight persons the mean MVPA is 3.397345 and standard deviation is 3.048872

For non- overweight persons the mean MVPA is 3.716766 and standard deviation is 2.639543

The total number of respondents is 300.

Thus df = 298.

Textbox 1: results of independent sample t-test

From the test results we find that p-value = 0.3653. Since p-value is more than0.05, level of significance. Hence, we reject Null Hypothesis.

Thus there is no difference in mean MVPA between overweight and non-overweight respondents.

Question 2

To analyse the research question the most appropriate test in the paired sample t-test. A paired sample t-test is used to analyse the difference in means of two variables which are related. In the present questions the respondent’s weight at enrolment and at graduation are compared. Since the variables weight at enrolment and at graduation is related through respondents hence the paired sample t-test / dependent sample t-test is used.

Question 3

Textbox 2: 95% CI of GPA of overweight and non-overweight students

The 95% confidence interval for the difference in mean GPA between overweight and non-overweight is 0.03989531, 0.46413733.

b) When another sample of the Australian university students is taken and the mean GPA of non-overweight students is 4.846707 and mean GPA of overweight students is 4.594690, then with 95% confidence it can be said that the mean difference of the sample would lie between 0.03989531 and 0.46413733.

c) From the calculation of 95% CI of mean GPA for overweight and non-overweight students it would be difficult to tell whether the results are statistically significant. Just from the results of 9% CI we do-not know the sample size, neither do we know the mean and standard deviation of the mean difference.

Question 4

The difference in means of the groups is 0.5.

The standard deviation of the groups = 0.9

Hence, the signal / noise ratio = 0.5/0.9 = 0.556

Thus, for signal / noise ratio of 0.556, with power = 0.8 and a = 0.05 the sample size required can be calculated as:

Textbox 3: Analysis of the required sample size

Thus the minimum sample size required in each group is 43 to detect a difference of 0.5 in mean GPA having standard deviation of 0.9, power = 0.8 and a = 0.05.

Question 5

Table 1: Crosstabulation of frequency of Gender vs Overweight

	Gender
Overweight	Male	Female
Not-overweight	98	69
Overweight	36	77

Table 2: Crosstabulation of row percentages of Gender vs Overweight

	Gender
Overweight	Male	Female	Total	Count
Not-overweight	58.7	41.3	100	167
Overweight	31.9	68.1	100	113

From looking at the table of frequency or percentage it would be difficult to tell if there is any relationship between gender and overweight.

From the tables we find that while the percentage of non-overweight males is higher than females, the percentage of males who are overweight is less than the percentage of females.

Similarly the percentage of Non-overweight males is higher than overweight males. On the other hand the percentage of non-overweight females is lower than overweight females.

The requirements of Chi-square tests are met –

Both the variables are nominal variables.
The sample size of both the groups under study is different
The distribution of the sample size is skewed.

Null Hypothesis: There is no association between Gender and OverWeight.

Alternate Hypothesis: There is an association between Gender and Overweight

Degrees of freedom = 1

Decision Rule : If the p-value is less than 0.05 then we accept the Null Hypothesis.

Textbox 4: Pearson’s Chi-square test

From the above textbox we find that c² = 19.433. The p-value is less than 0.001 at a=0.05. Hence we accept the Null Hypothesis. Thus there is no association between Gender and Over weight

Question 6

The sample proportion that is overweight = 0.4

The margin of error = 0.1

The confidence interval = 0.95%

Textbox 5: Calculation for sample size

Thus for a population proportion of 40% at 95% CI a minimum sample size of 93 is required to achieve a 10% margin of error for the survey of overweight Australia university students.