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Chapter 2: Problem 11

A group of laboratory animals is infected with a particular form of bacteria, and their survival time is found to average 32 days, with a standard deviation of 36 days. a. Visualize the distribution of survival times. Do you think that the distribution is relatively mound shaped, skewed right, or skewed left? Explain. b. Within what limits would you expect at least \(3 / 4\) of the measurements to lie?

Short Answer

Expert verified
Answer: The approximate limits within which at least 3/4 of the survival time measurements lie are between 0 days and 68 days.

Step by step solution

01

Understanding the given information

A group of laboratory animals is infected with a particular form of bacteria, and their survival time has an average of 32 days, with a standard deviation of 36 days. This tells us that the data follows a distribution with a mean (\(\mu\)) of 32 days and a standard deviation (\(\sigma\)) of 36 days.
02

Visualize the distribution of survival times

Since no dataset is provided for the survival times, we cannot create a direct graphical representation of the distribution. However, we can discuss the possible shapes of the distribution, which include: - Mound-shaped or symmetric: The distribution is symmetric around the mean, with most of the data concentrated towards the center. - Skewed right: The distribution is asymmetrical, with a longer right tail indicating a larger number of observations towards higher survival times. - Skewed left: The distribution is asymmetrical, with a longer left tail indicating a larger number of observations towards lower survival times. Without actual data values or a histogram, we cannot definitively determine the shape of the distribution in this exercise.
03

Calculating the limits for 3/4 of the measurements

To find the limits within which at least 3/4 of the measurements lie, we can use the empirical rule for normal distribution (assuming normality). According to the empirical rule, approximately 68% of the data lie within one standard deviation of the mean, approximately 95% lie within two standard deviations, and 99.7% lie within three standard deviations. To find the limits for 3/4 (75%) of the measurements, we will estimate the value between 68% and 95%: 1. Lower limit: Mean - Standard Deviation = \(\mu - \sigma\) = \(32 - 36\) = \(-4\) 2. Upper limit: Mean + Standard Deviation = \(\mu + \sigma\) = \(32 + 36\) = \(68\) So, the limits within which at least 3/4 of the measurements lie are approximately (\(-4\) days, \(68\) days). Keep in mind that since the survival time cannot be negative, the lower limit in practice would be \(0\) days.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Survival Analysis
Survival Analysis is a statistical method used to analyze the expected duration of time until one or more events happen, such as death in biological organisms. It's often applied in clinical trials and health research, where understanding time-to-event data is crucial. In the context of our example, survival analysis helps us understand the distribution and spread of survival times for a group of infected animals. By assessing the mean survival time and its variability, we can predict survival trends over a specified period.
  • Uses: It helps in identifying the probability of survival over a certain period.
  • Techniques: Often uses Kaplan-Meier curves and Cox proportional hazards models.
Knowing these techniques allows researchers to plan better interventions and treatment strategies to improve survival outcomes.
Empirical Rule
The Empirical Rule is a guideline that helps us understand the spread of data in a normal distribution. It's sometimes referred to as the "68-95-99.7 Rule," referring to the percentage of data expected to lie within one, two, and three standard deviations from the mean.
  • One standard deviation from the mean captures about 68% of the data.
  • Two standard deviations capture about 95%.
  • Three standard deviations capture about 99.7%.
In the exercise, we use this rule to estimate the range within which a significant portion of the laboratory animals' survival times will fall. Although the actual data distribution isn't given, assuming a normal distribution allows us to apply the empirical rule for a good estimation.
Data Distribution
Data Distribution refers to how values in a data set spread out across their range. Understanding the shape of the distribution 鈥� whether it is normal, skewed, or otherwise 鈥� is crucial in statistics, as it influences how data should be approached and interpreted.
  • Mound-shaped (normal): Data is symmetrically distributed around the mean.
  • Skewed right: More low values, with the tail extending to the higher side.
  • Skewed left: More high values, with the tail extending to the lower side.
In our example, we lacked detailed data, so we speculated on the distribution shape based on the given mean and standard deviation. This helps in understanding the variability and expectations for survival times.
Standard Deviation
Standard Deviation is a measure that indicates the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean of the set, whereas a high standard deviation indicates that the values are spread out over a wider range.
  • Formula: Given by \( \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2} \), where \( \mu \) is the mean and \( x \) are the data points.
  • Role in prediction: It provides a quantitative measure for predicting outcomes.
  • In the exercise: A high standard deviation of 36 days suggests significant variability in survival times.
Understanding standard deviation allows researchers to assess whether their data points are typical or atypical compared to the average, aiding in decision making.

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Most popular questions from this chapter

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