What was the age distribution of nurses in Great Britain at...

What was the age distribution of nurses in Great Britain at the time of Florence Nightingale? Thanks to Florence Nightingale and the British census of 1851 , we have the following information (based on data from the classic text Notes on Nursing, by Florence Nightingale). Note: In 1851 there were 25,466 nurses in Great Britain. Furthermore, Nightingale made a strict distinction between nurses and domestic servants. $$ \begin{array}{l|ccccccc} \hline \text { Age range (yr) } & 20-29 & 30-39 & 40-49 & 50-59 & 60-69 & 70-79 & 80+ \\ \hline \text { Midpoint } x & 24.5 & 34.5 & 44.5 & 54.5 & 64.5 & 74.5 & 84.5 \\ \hline \begin{array}{l} \text { Percent of } \\ \text { nurses } \end{array} & 5.7 \% & 9.7 \% & 19.5 \% & 29.2 \% & 25.0 \% & 9.1 \% & 1.8 \% \\ \hline \end{array} $$ (a) Using the age midpoints $x$ and the percent of nurses, do we have a valid probability distribution? Explain. (b) Use a histogram to graph the probability distribution of part (a). (c) Find the probability that a British nurse selected at random in 1851 was 60 years of age or older. (d) Compute the expected age $\mu$ of a British nurse contemporary to Florence Nightingale. (e) Compute the standard deviation $\sigma$ for ages of nurses shown in the distribution.

What was the age distribution of nurses in Great Britain at the time of Florence Nightingale? Thanks to Florence Nightingale and the British census of 1851 , we have the following information (based on data from the classic text Notes on Nursing, by Florence Nightingale). Note: In 1851 there were 25,466 nurses in Great Britain. Furthermore, Nightingale made a strict distinction between nurses and domestic servants.
$$
\begin{array}{l|ccccccc}
\hline \text { Age range (yr) } & 20-29 & 30-39 & 40-49 & 50-59 & 60-69 & 70-79 & 80+ \\
\hline \text { Midpoint } x & 24.5 & 34.5 & 44.5 & 54.5 & 64.5 & 74.5 & 84.5 \\
\hline \begin{array}{l}
\text { Percent of } \\
\text { nurses }
\end{array} & 5.7 \% & 9.7 \% & 19.5 \% & 29.2 \% & 25.0 \% & 9.1 \% & 1.8 \% \\
\hline
\end{array}
$$
(a) Using the age midpoints $x$ and the percent of nurses, do we have a valid probability distribution? Explain.
(b) Use a histogram to graph the probability distribution of part (a).
(c) Find the probability that a British nurse selected at random in 1851 was 60 years of age or older.
(d) Compute the expected age $\mu$ of a British nurse contemporary to Florence Nightingale.
(e) Compute the standard deviation $\sigma$ for ages of nurses shown in the distribution.

Key Concepts

Probability Distribution

A probability distribution describes how the probabilities are assigned to possible outcomes or intervals. In this context, it involves assessing whether the percentages (when properly normalized) sum up to 100% or 1, ensuring that the distribution is valid and adheres to the fundamental rules of probability.

Histogram

A histogram is a graphical tool that represents the frequency or probability of data falling within specified intervals, or bins. It is used to visualize the shape, spread, and central tendency of a data distribution, making it easier to interpret how the underlying data is distributed across different age groups.

Expected Value (Weighted Average)

The expected value, or weighted average, of a probability distribution is computed by multiplying each outcome by its probability and summing all these products. This measure provides a central value or 'mean' of the distribution that reflects the long-term average outcome over many observations.

Standard Deviation

Standard deviation is a measure of dispersion that indicates how data values spread out from the mean. A larger standard deviation suggests greater variability among the outcomes, whereas a smaller standard deviation indicates that the data points are closer to the mean, providing insight into the consistency of the distribution.

Cumulative Probability

Cumulative probability involves summing probabilities of outcomes up to a certain point or threshold, which is useful for determining the likelihood of a variable falling within a specific range. This concept is particularly important when calculating the probability that an outcome meets or exceeds a given criterion, such as age or value thresholds.

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