Autumn 2019 Biostatistics Assignment 2 - Statistical Analysis

This document presents a complete solution to a biostatistics assignment. The assignment focuses on analyzing a dataset related to grandparent carers, exploring concepts such as confidence intervals, hypothesis testing, and statistical tests. The solution includes the calculation and interpretation of a 95% confidence interval for mean grip strength, and a determination if the mean grip strength is significantly different from a specified value. The document tests hypotheses comparing mean grip strength between males and females, and investigates the proportion of grandparent carers with hypertension. Additionally, the solution involves a Wilcoxon signed-rank test and sample size calculations for various scenarios. All answers are supported by statistical justifications and relevant references.

401077 Introduction to Biostatistics, Autumn 2019
Assignment 2
Due Sunday September 15, 2019
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Question 1
a) The 95% confidence interval of mean grip strength is (31.35763, 32.36045). With 95%
confidence, the true mean of grip strength lies between 31.35763 and 32.36045.
b) The 95% confidence interval of mean grip strength confirms that the mean value is not
significantly equal to 33 kgs at the 0.05 significance level.
The mean of grip strength that is 33 kgs does not lie between 31.35763 and 32.36045 hence
it value is not significantly equal to 33 kgs at the 0.05 significance level.
c) The Null Hypothesis (H0): There exists a difference between male and female mean grip
strength.
Alternative Hypothesis (HA): There does not exist a difference between male and female mean grip
strength.

The Significance Level: The significance level is set at 0.05.
The Test Statistic and Corresponding P-Value: The male and female mean grip strength is 31.60399
and 32.11191 respectively. Now, the estimated t-stat is -0.99822 with 0.3192 p-value that is greater
than 0.05 (Heimann et al., 2019). The p-value suggests not to reject the null hypothesis.
Conclusion: The mean grip strength does not differ between male and female.
d) The Null Hypothesis (H0): There is no shift in location.
Alternative Hypothesis (HA): There is no shift in location.
The Significance Level: The significance level is set at 0.05.
The Test Statistic and Corresponding P-Value: w-stat is 6418 with 0.475 p-value which is greater
than 0.05. The p-value suggests to retain the null hypothesis.
Drawing a Conclusion: The location shift is equal to zero.
Question 2
a) The Null Hypothesis (H0): The proportion of grandparent carers with hypertension is not
greater than 0.25.
Alternative Hypothesis (HA): The proportion of grandparent carers with hypertension is greater than
0.25.
The Significance Level: The significance level is set at 0.05.
The Test Statistic and Corresponding P-Value: The χ2 stat is 2.4049 with 1 df (Pagano & Gauvreau,
2018). The p-value is 0.06048 greater than 0.05. Hence, the null hypothesis is retained.
Drawing a Conclusion: The proportion of grandparent carers with hypertension is not greater than
0.25.
b) The 95% confidence interval of the difference between male and female proportion
with hypertension is (0.03286422, 0.09589114). The difference between male and female
proportion with hypertension will lie between 0.03286422 and 0.09589114 with 95%
confidence.
Question 3
Table 1: Willcoxon Signed Rank Test

participant Dominant hand Grip
strength (kgs)
Non-dominant
hand Grip
strength (kgs)
differenc
e Positive |Diff| Rank Signed
Rank
1 27 13 14 1 14 8 8
2 30 28 2 1 2 3 3
3 36 30 6 1 6 6 6
4 31 30 1 1 1 1 1
5 39 38 1 1 1 1 1
6 33 29 4 1 4 5 5
7 35 32 3 1 3 4 4
8 22 29 -7 -1 7 7 -7
The positive sum is 28.
The negative sum is -7.
The minimum or the test statistics is 7.
The critical value is 6.
The Null Hypothesis (H0): The grip strength of non-dominant hands is same to the grip
strength of dominant hand.
Alternative Hypothesis (HA): The grip strength of non-dominant hands is lower than the grip
strength of dominant hand.
The Significance Level: The significance level is set at 0.05.
The Test Statistic: The test statistics of the test is 7. The critical value of test statistic for one
tailed test is 6 (Couch et al., 2018). The test statistics is greater than critical value. Hence null
hypothesis is rejected.
Drawing a Conclusion: The grip strength of non-dominant hands is lower than the grip
strength of dominant hand.
Question 4
a) The proportion of female with hypertension is equal to 0.1158798.
b) The minimum sample size is 24 which is needed to test the clinically significant
difference between male and females with hypertension, with 80% power at the α =0.05 .

c) The standard deviation of grip strength is 3.884621.
d) The minimum sample size is 26 that is required to produce a 95% confidence interval
of the mean of grip strength with a margin error of 1.5 kgs (Machin et al., 2018).
e) The confidence will be lower if the confidence interval is set at 50% instead of 95%.

Reference
Couch, S., Kazan, Z., Shi, K., Bray, A., & Groce, A. (2018). A Differentially Private Wilcoxon
Signed-Rank Test. arXiv preprint arXiv:1809.01635.
Heimann, S. M., Vehreschild, M. J., Cornely, O. A., Heinz, W. J., Grüner, B., Silling, G., ... &
Vehreschild, J. J. (2019). Healthcare burden of probable and proven invasive
mucormycosis: a multi-centre cost-of-illness analysis of patients treated in tertiary
care hospitals between 2003 and 2016. Journal of Hospital Infection, 101(3), 339-
346.
Machin, D., Campbell, M. J., Tan, S. B., & Tan, S. H. (2018). Sample Sizes for Clinical,
Laboratory and Epidemiology Studies. Wiley-Blackwell.
Pagano, M., & Gauvreau, K. (2018). Principles of biostatistics. Chapman and Hall/CRC.