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Chapter 4: Q93E (page 193)

Construct a probability plot that will allow you to assess the plausibility of the lognormal distribution as a model for the rainfall data.

Short Answer

Expert verified

Yes, the rainfall data is likely to have a lognormal distribution.

Step by step solution

01

Definition

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

02

Construct a probability plot

The Lognormal probability plot is the plot of \({\rm{ln}}\)(observation) against their corresponding z-percentiles. To calculate the percentiles, we first need to calculate \({{\rm{p}}_{\rm{i}}}\)values.

\({{\rm{p}}_{\rm{i}}}{\rm{ = }}\frac{{{\rm{i - 0}}{\rm{.5}}}}{{\rm{n}}}\)

Where \({\rm{n}}\) denotes the total number of sample values. Then z-percentile corresponding to the \({{\rm{i}}^{{\rm{th }}}}\) sample value is the \({\left( {{\rm{100}}{{\rm{p}}_{\rm{i}}}} \right)^{{\rm{th }}}}\)percentile of standard normal distribution curve. Or in other words, they are z-scores corresponding to the probability \({{\rm{p}}_{\rm{i}}}\) (Use Appendix \({\rm{A - 3}}\)or use software).

The table below lists sample values \({\rm{(x)}}\)and their corresponding\({\rm{ln(x),p}}\)values and z-percentiles.

03

Construct the graph

The matching Lognormal probability curve is shown below. The plot's pattern is fairly straight, indicating that the rainfall data is likely to have a lognormal distribution.

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Most popular questions from this chapter

There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean \({\bf{3}}\) cm and standard deviation \(.{\bf{1}}\) cm. The second machine produces corks with diameters that have a normal distribution with mean \({\bf{3}}.{\bf{04}}\)cm and standard deviation \(.{\bf{02}}\)cm. Acceptable corks have diameters between \({\bf{2}}.{\bf{9}}\)cm and \({\bf{3}}.{\bf{1}}\)cm. Which machine is more likely to produce an acceptable cork?

Two machines that produce wine corks, the first one having a normal diameter distribution with mean value \({\rm{3\;cm}}\) and standard deviation\({\rm{.1\;cm}}\), and the second having a normal diameter distribution with mean value \({\rm{3}}{\rm{.04\;cm}}\) and standard deviation\({\rm{.02\;cm}}\). Acceptable corks have diameters between \({\rm{2}}{\rm{.9}}\) and\({\rm{3}}{\rm{.1\;cm}}\). If \({\rm{60\% }}\) of all corks used come from the first machine and a randomly selected cork is found to be acceptable, what is the probability that it was produced by the first machine?

If \({\rm{X}}\) has an exponential distribution with parameter \({\rm{\lambda }}\), derive a general expression for the \({\rm{(100p)}}\)th percentile of the distribution. Then specialize to obtain the median.

Let \({{\rm{I}}_{\rm{i}}}\) be the input current to a transistor and \({{\rm{I}}_{\rm{0}}}\) be the output current. Then the current gain is proportional to\({\rm{ln}}\left( {{{\rm{I}}_{\rm{0}}}{\rm{/}}{{\rm{I}}_{\rm{i}}}} \right)\). Suppose the constant of proportionality is \({\rm{1}}\) (which amounts to choosing a particular unit of measurement), so that current gain\({\rm{ = X = ln}}\left( {{{\rm{I}}_{\rm{0}}}{\rm{/}}{{\rm{I}}_{\rm{i}}}} \right)\). Assume \({\rm{X}}\) is normally distributed with \({\rm{\mu = 1}}\) and\({\rm{\sigma = }}{\rm{.05}}\).

a. What type of distribution does the ratio \({{\rm{I}}_{\rm{0}}}{\rm{/}}{{\rm{I}}_{\rm{i}}}\) have?

b. What is the probability that the output current is more than twice the input current?

c. What are the expected value and variance of the ratio of output to input current?

Let X denote the number of flaws along a \({\bf{100}}\)-m reel of magnetic tape (an integer-valued variable). Suppose X has approximately a normal distribution with m \(\mu = 25\) and s \(\sigma = 5\). Use the continuity correction to calculate the probability that the number of flaws is

a. Between \({\bf{20}}\) and \({\bf{30}}\), inclusive.

b. At most \({\bf{30}}\). Less than \({\bf{30}}\).
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