Q1 The Following Data Shows The Liver Sizes Measured To T

Q1the Following Data Shows The Liver Sizes Measured To T

The assignment involves analyzing data related to biological measurements and conducting statistical analyses. Specifically, it requires calculating descriptive statistics for liver sizes measured in chimpanzees, constructing confidence intervals for pulmonary function metrics, performing hypothesis tests comparing drug effects, analyzing the probability of reactions to Tamiflu, and evaluating blood pressure data among men, as well as comparing snake snout-vent lengths in captive and wild populations. These tasks involve applying principles of descriptive statistics, inferential statistics, probability, normal distribution, confidence intervals, sample size determination, and hypothesis testing based on sample data.

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The statistical analyses outlined in the assignment encompass key concepts in descriptive and inferential statistics, probability theory, and hypothesis testing. The first part involves summarizing liver size data measured in 40 chimpanzees, requiring calculations of the mean, median, mode, standard deviation, and quartile deviation to describe the data's distribution and central tendency. These statistics provide insight into the typical liver size and variability within the sample, crucial for understanding biological variation in primates.

In the second part, the focus shifts to pulmonary function data from 144 individuals with exercise-induced asthma. Constructing a 95% confidence interval for the mean forced vital capacity (FVC) involves applying the formula for confidence bounds, which incorporates the sample mean, standard deviation, and the critical value from the t-distribution. Estimating the sample size needed to reduce the interval width to 0.1 liters requires manipulating the confidence interval formula and solving for the sample size, assuming the population mean is known. Subsequently, a hypothesis test compares the sample mean against the known population mean of 2.65 liters at a 1% significance level. The null hypothesis posits no reduction in FVC for asthma sufferers; rejection indicates significant evidence of reduced capacity.

The third segment explores the proportion of patients suffering side effects from two drugs used to treat hay fever. For drug A, a sample of 300 patients yielded 45 side effects. Calculating a 95% confidence interval for this proportion employs the normal approximation to the binomial distribution. Estimating additional patients needed to narrow the interval to 0.05 involves solving for the sample size required to achieve this precision. A comparison between drugs A and B using a 1% significance level involves testing whether the difference in side effect proportions is statistically significant, often through a two-proportion z-test.

In the fourth section, the probability of reactions to Tamiflu is modeled using the binomial distribution with p=0.05. Calculations include finding the probability of exactly two reactions among 20 patients and more than five reactions, utilizing binomial probability formulas or normal approximation when applicable. For larger samples, the mean and variance of the number of reactions are computed based on the binomial parameters. Applying the normal approximation, the probability that between 12 and 15 reactions occur among 500 patients is estimated, demonstrating the use of central limit theorem for large sample sizes.

The fifth component involves analyzing diastolic blood pressure data assumed normally distributed with mean 82 mmHg and standard deviation 14.5 mmHg in Icelandic men aged 40 to 75. Calculating probabilities for blood pressure within specified ranges employs the standard normal distribution. For a sample of 50 men, the distribution of the sample mean diastolic pressure is derived, and the probability that this mean is below 81.7 mmHg is computed. Determining the required sample size to reduce the probability of a sample mean below 81.7 mmHg to 0.1 involves applying the properties of the sampling distribution and solving for the sample size.

The final part concerns the development of snakes, measured through snout-vent length. For snakes bred in captivity, the sample mean and standard deviation are computed, and a 95% confidence interval for the population mean length is constructed. Comparing these lengths with those from wild-bred snakes, with known sample mean and standard deviation, involves hypothesis testing at a 1% significance level to determine if a significant difference exists between the two populations. These analyses integrate inferential statistics and hypothesis testing principles, essential for understanding biological variation and assessing treatment or breeding effects.

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