Question

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

Short answer

Based on the given average (mean) and standard deviation, we cannot definitively determine the shape of the distribution of survival times without more information or a graphical representation of the data. However, by using Chebyshev's theorem, we can find the limits within which at least 3/4 of the measurements would lie. These limits are 0 days (since survival time cannot be negative) and 104 days.

Step-by-step solution

(Exercise A: Determining the Distribution's Shape)

To determine the shape of the distribution, we may rely on the mean and standard deviation provided. However, we don't have enough information to definitively say if the distribution is mound-shaped, skewed right or skewed left. An easier approach would be to analyze the data graphically to determine the distribution. Without a histogram or box plot, we can't give a definite answer.

(Exercise B: Finding the Limits for 3/4 of Measurements)

To find the limits within which 3/4 of the measurements lie, we can use Chebyshev's theorem. Chebyshev's theorem states that at least \(1-\frac{1}{k^2}\) of measurements will lie within k standard deviations (SD) of the mean for any dataset. In our case, we want to find the limits for at least \(3/4 = 0.75 = 1-\frac{1}{k^2}\) of the measurements. First, let's find k. Solve the equation: \(1-\frac{1}{k^2} = 0.75\) \(0.25 = \frac{1}{k^2}\) \(k^2 = 4\) \(k = \sqrt{4} = 2\) Now, to find the limits, we'll calculate the lower and upper limits using the mean and standard deviation. Mean = 32 days Standard deviation = 36 days Lower limit: Mean - k * Standard deviation Lower limit: \(32 - 2 \times 36 = 32 - 72 = -40\) Upper limit: Mean + k * Standard deviation Upper limit: \(32 + 2 \times 36 = 32 + 72 = 104\) So, at least 3/4 of the measurements would lie within the limits of -40 days to 104 days. Keep in mind that survival time cannot be negative so practically, the lower limit would be 0 days, and the bounds would be between 0 to 104 days.

Key concepts

Distribution of Survival Times

When studying survival analysis, it's essential to understand the distribution of survival times, which refers to the pattern observed in the lengths of time until events of interest occur, such as death in biological organisms or failure in mechanical systems. In this context, time is the random variable of interest, and its probability distribution characterizes the likelihood of survival for different time intervals.

Common distributions used in survival analysis include the exponential, Weibull, and log-normal distributions. Each has different properties regarding how the events are spread over time. For instance, if survival times tend to cluster around a mean with few extreme values, the distribution might be mound-shaped or symmetric. Conversely, if there are more extended survival times that tail off to the right or shorter times tailing to the left, the distribution could be skewed right or left, respectively.

In the given exercise, without graphical representations like histograms or box plots, it's challenging to ascertain the exact shape purely based on the mean and standard deviation. Hence, we consider the context and use additional data or plots to make a reliable assumption about the distribution of survival times.

Chebyshev's Theorem

Chebyshev's theorem is a powerful statistical rule that applies to all data sets, regardless of the underlying distribution. It gives us a way to state how much of the data falls within a certain number of standard deviations from the mean.

The theorem states that for any dataset and for any real number k greater than 1, at least \(1-1/k^2\) of the data values will be within k standard deviations of the mean. This theorem provides a way to estimate the spread of the data without making any assumptions about the data's distribution.

Application in Exercise

For the exercise, where we want to enclose at least \({3}/{4}\) of the survival times, the theorem helps us determine that at least 75% will lie within two standard deviations (\(k=2\)). Although real-life survival times can't be negative, the theorem itself doesn't account for the nature of the data. It's a purely mathematical estimate of spread, hence, the practical adjustment to the lower limit to start from 0.

Standard Deviation

Standard deviation is a measure that quantifies 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 (average) of the set, while a high standard deviation means that the values are spread out over a broader range.

In statistics, standard deviation plays a vital role as it gives context to the mean. For example, knowing that survival times average 32 days tells us little unless we have an idea of the variability of those times, which is where standard deviation comes into the picture.

In our exercise, a standard deviation of 36 days relative to the mean of 32 days indicates a significant spread in the data. This implies that the survival times are quite spread out around the mean, and there can be a wide range of survival times observed within the group of laboratory animals. When calculating expected ranges for survival times, we must incorporate this variability to provide meaningful predictions.