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Chapter 5: Q43E (page 230)

Suppose the amount of liquid dispensed by a certain machine is uniformly distributed with lower limit \({\rm{A = 8oz}}\) and upper limit\({\rm{B = 10oz}}\). Describe how you would carry out simulation experiments to compare the sampling distribution of the (sample) fourth spread for sample sizes\({\rm{n = 5,10,20}}\), and\({\rm{30}}\).

Short Answer

Expert verified

This yields \({\rm{1000}}\) fourth spreads for each sample size. To compare the sample distributions, use any method to visualize the data. Compare the results of each sample distribution on its own histogram.

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

Determining how would carry out simulation experiments

Assume we have a total of \({\rm{n}}\) observations. We define the low fourth as the median of the smallest half and the upper fourth as the median of the biggest half by sorting the observations from smallest to largest and splitting them into two halves (smallest half/largest half; if \({\rm{n}}\) is odd, the median \({\rm{\tilde x}}\) is included in both halves). The fourth spread, \({{\rm{f}}_{\rm{s}}}\), is defined as follows:

\({{\rm{f}}_{\rm{s}}}{\rm{ = upper fourth - lower fourth}}{\rm{. }}\)

The fourth spread contains the statistic of interest. It is simple to calculate the value of the fourth spread given data. We must first simulate the data in order to compare the sampling distribution of the fourth spread for specified sample sizes \({\rm{n = 5,10,20,40}}\).

For example, using a uniform distribution with stated bounds \({\rm{A}}\) and\({\rm{B}}\), use a computer to create samples of the specified sizes. Take the same number of replications for each sample (it does not have to be really big number, \({\rm{1000}}\)should be enough).

Calculate the fourth spread for each replication as follows:

\({{\rm{f}}_{\rm{s}}}{\rm{ = upper fourth - lower fourth}}{\rm{. }}\)

For each sample size, this results in fourth spreads. Use any method to visualize the data to compare the sample distributions. For example, make a distinct histogram for each sample distribution and compare them.

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Most popular questions from this chapter

There are two traffic lights on a commuter's route to and from work. Let \({{\rm{X}}_{\rm{1}}}\) be the number of lights at which the commuter must stop on his way to work, and \({{\rm{X}}_{\rm{2}}}\) be the number of lights at which he must stop when returning from work. Suppose these two variables are independent, each with pmf given in the accompanying table (so \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\) is a random sample of size \({\rm{n = 2}}\)).

a. Determine the pmf of \({{\rm{T}}_{\rm{o}}}{\rm{ = }}{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}\).

b. Calculate \({{\rm{\mu }}_{{{\rm{T}}_{\rm{o}}}}}\). How does it relate to \({\rm{\mu }}\), the population mean?

c. Calculate \({\rm{\sigma }}_{{{\rm{T}}_{\rm{o}}}}^{\rm{2}}\). How does it relate to \({{\rm{\sigma }}^{\rm{2}}}\), the population variance?

d. Let \({{\rm{X}}_{\rm{3}}}\) and \({{\rm{X}}_{\rm{4}}}\) be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the first day. With \({{\rm{T}}_{\rm{o}}}{\rm{ = }}\) the sum of all four \({{\rm{X}}_{\rm{i}}}\) 's, what now are the values of \({\rm{E}}\left( {{{\rm{T}}_{\rm{a}}}} \right)\) and \({\rm{V}}\left( {{{\rm{T}}_{\rm{a}}}} \right)\)?

e. Referring back to (d), what are the values of \({\rm{P}}\left( {{{\rm{T}}_{\rm{o}}}{\rm{ = 8}}} \right)\) and \(\text{P}\left( {{\text{T}}_{\text{e}}}\text{ }\!\!{}^\text{3}\!\!\text{ 7} \right)\) (Hint: Don't even think of listing all possible outcomes!)

a. Compute the covariance between \({\rm{X}}\) and\({\rm{Y}}\).

b. Compute the correlation coefficient \({\rm{\rho }}\) for this \({\rm{X}}\) and \({\rm{Y}}\).

Young鈥檚 modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are \({\rm{70 GPa}}\) and \({\rm{1}}{\rm{.6 GPa}}\), respectively (values given in the article 鈥淚nfluence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis鈥� (Intl. J. of Advanced Manuf. Tech., \({\rm{2010:117 - 134}}\))).

  1. If \({\rm{\bar X}}\) is the sample mean young鈥檚 modulus for a random sample of \({\rm{n = 16}}\)sheets, where is the sampling distribution of \({\rm{\bar X}}\)centered, and what is the standard deviation of the \({\rm{\bar X}}\)distribution?
  2. Answer the questions posed in part (a) for a sample size of \({\rm{n = 64}}\)sheets.
  3. For which of the two random samples, the one of part (a) or the one of part (b), is \({\rm{\bar X}}\) more likely to be within \({\rm{1GPa}}\) of \({\rm{70 GPa}}\)? Explain your reasoning.

Suppose the proportion of rural voters in a certain state who favor a particular gubernatorial candidate is\(.{\bf{45}}\)and the proportion of suburban and urban voters favouring the candidate is\(.{\bf{60}}\). If a sample of\({\bf{200}}\)rural voters and\({\bf{300}}\)urban and suburban voters is obtained, what is the approximate probability that at least\(\;{\bf{250}}\)of these voters favour this candidate?

A box contains ten sealed envelopes numbered\({\rm{1, \ldots ,10}}\). The first five contain no money, the next three each contains\({\rm{\$ 5}}\), and there is a \({\rm{\$ 10}}\) bill in each of the last two. A sample of size \({\rm{3}}\) is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. If \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and \({{\rm{X}}_{\rm{3}}}\) denote the amounts in the selected envelopes, the statistic of interest is \({\rm{M = }}\) the maximum of\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and\({{\rm{X}}_{\rm{3}}}\).

a. Obtain the probability distribution of this statistic.

b. Describe how you would carry out a simulation experiment to compare the distributions of \({\rm{M}}\) for various sample sizes. How would you guess the distribution would change as \({\rm{n}}\) increases?
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