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Chapter 5: Q48E (page 237)

The National Health Statistics Reports dated Oct. \({\rm{22, 2008}}\), stated that for a sample size of \({\rm{277 18 - }}\)year-old American males, the sample mean waist circumference was \({\rm{86}}{\rm{.3cm}}\). A somewhat complicated method was used to estimate various population percentiles, resulting in the following values:

a. Is it plausible that the waist size distribution is at least approximately normal? Explain your reasoning. If your answer is no, conjecture the shape of the population distribution.

b. Suppose that the population mean waist size is \({\rm{85cm}}\)and that the population standard deviation is \({\rm{15cm}}\). How likely is it that a random sample of \({\rm{277}}\) individuals will result in a sample mean waist size of at least \({\rm{86}}{\rm{.3cm}}\)?

c. Referring back to (b), suppose now that the population mean waist size in \({\rm{82cm}}\).Now what is the (approximate) probability that the sample mean will be at least \({\rm{86}}{\rm{.3cm}}\)? In light of this calculation, do you think that \({\rm{82cm}}\)is a reasonable value for \({\rm{\mu }}\)?

Short Answer

Expert verified

a. No, Right-skewed

b. \({\rm{7}}{\rm{.49}}\)percent likely

d. Less than \({\rm{0}}{\rm{.0001}}\) (\({\rm{0}}{\rm{.01\% }}\)), No

Step by step solution

01

Definition of standard deviation

The square root of the variance is the standard deviation of a random variable, sample, statistical population, data collection, or probability distribution. It is less resilient in practice than the average absolute deviation, but it is algebraically easier.

02

Calculating (a);

Provided:

\({\rm{\bar x = 86}}{\rm{.3}}\)

The median \({\rm{81}}{\rm{.3}}\), and the \({\rm{50th}}\) percentile is \({\rm{81}}{\rm{.3cm}}\)

The sample mean of \({\rm{86}}{\rm{.3cm}}\)is higher than the sample median of \({\rm{81}}{\rm{.3cm}}\) (though still within the \({\rm{25th}}\) and \({\rm{75th}}\) percentiles), indicating that the distribution is right-skewed rather than essentially normal.

03

calculating (b);

Provided:

\(\begin{array}{*{20}{c}}{{\rm{\bar x = 86}}{\rm{.3}}}\\{{\rm{\mu = 85}}}\\{{\rm{\sigma = 15}}}\\{{\rm{n = 277}}}\end{array}\)

The sample mean's sampling distribution has a mean of \({\rm{\mu }}\)and a standard deviation of \(\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}\)

The \({\rm{z}}\)-score is the difference between the population mean and the standard deviation divided by the population mean:

\({\rm{z = }}\frac{{{\rm{\bar x - \mu }}}}{{{\rm{\sigma /}}\sqrt {\rm{n}} }}{\rm{ = }}\frac{{{\rm{86}}{\rm{.3 - 85}}}}{{{\rm{15/}}\sqrt {{\rm{277}}} }}{\rm{\gg 1}}{\rm{.44}}\)

Using a normal probability table, calculate the equivalent probability:

\(P(\bar x^{3}86.3) = P(z > 1.44) = 1 - P(z < 1.44) = 1 - 0.9251 = 0.0749 = 7.49\% \)

The event is likely to occur since the likelihood is greater than \({\rm{5\% }}\)

04

Calculating (c);

Provided:

\(\begin{array}{*{20}{c}}{{\rm{\bar x = 86}}{\rm{.3}}}\\{{\rm{\mu = 85}}}\\{{\rm{\sigma = 15}}}\\{{\rm{n = 277}}}\end{array}\)

The sample mean's sampling distribution has a mean of \({\rm{\mu }}\)and a standard deviation of \(\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}\)

The \({\rm{z}}\)-score is the difference between the population mean and the standard deviation divided by the population mean:

\({\rm{z = }}\frac{{{\rm{\bar x - \mu }}}}{{{\rm{\sigma /}}\sqrt {\rm{n}} }}{\rm{ = }}\frac{{{\rm{86}}{\rm{.3 - 82}}}}{{{\rm{15/}}\sqrt {{\rm{277}}} }}{\rm{\gg 4}}{\rm{.77}}\)

Using a normal probability table, calculate the equivalent probability:

\(P(\bar x^{3}86.3) = P(z > 4.77) = 1 - P(z < 4.77) < 1 - 0.9999 = 0.0001 = 0.01\% \)

\(82cm\) Because the probability is nearly \({\rm{0}}{\rm{.}}\)is NOT a suitable value for \({\rm{\mu }}\)

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Most popular questions from this chapter

The lifetime of a certain type of battery is normally distributed with mean value \({\rm{10}}\)hours and standard deviation \({\rm{1}}\)hour. There are four batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only \({\rm{5\% }}\)of all packages?

The joint probability di\({\rm{Y}} \le {\rm{1) = 0}}{\rm{.12}}\)stribution of the number \({\rm{X}}\) of cars and the number \({\rm{Y}}\) of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table.

a. What is the probability that there is exactly one car and exactly one bus during a cycle?

b. What is the probability that there is at most one car and at most one bus during a cycle?

c. What is the probability that there is exactly one car during a cycle? Exactly one bus?

d. Suppose the left-turn lane is to have a capacity of five cars, and that one bus is equivalent to three cars. What is t\({\rm{p(x,y)}} \ge {\rm{0}}\)e probability of an overflow during a cycle?

e. Are \({\rm{X}}\) and \({\rm{Y}}\) independent rv鈥檚? Explain.

Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean \({\rm{2}}{\rm{.65 }}\)and standard deviation \({\rm{.85}}\) (suggested in 鈥淢odeling Sediment and Water Column Interactions for Hydrophobic Pollutants,鈥� Water Research, \({\rm{1984: 1164 - 1174 }}\)).

a. If a random sample of \({\rm{25}}\)specimens is selected, what is the probability that the sample average sediment density is at most \({\rm{3}}{\rm{.00 }}\)? Between \({\rm{2}}{\rm{.65 }}\)and \({\rm{3}}{\rm{.00 }}\)?

b. How large a sample size would be required to ensure that the first probability in part (a) is at least \({\rm{.99}}\)?

Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv鈥檚 with expected values \({\mu _1}, {\mu _2}, and {\mu _3}\)and variances \(\sigma _1^2 , \sigma _2^2, and \sigma _3^2 \), respectively. a. If \(\mu = {\mu _2} = {\mu _3} = 60\)and\(\sigma _1^2 = \sigma _2^2 = \sigma _3^2 = 15\), calculate \(P\left( {{T_0} \le 200} \right)\)and\(P\left( {150 \le {T_0} \le 200} \right)\)? b. Using the \(\mu 's and \sigma 's\)given in part (a), calculate both \(P\left( {55 \le X} \right)\)and \(P\left( {58 \le X \le 62} \right)\).c. Using the \(\mu 's and \sigma 's\)given in part (a), calculate and interpret\(P\left( { - 10 \le {X_1} - .5{X_2} - .5{X_3} \le 5} \right)\). d. If\({\mu _1} = 40, {\mu _1} = 50, {\mu _1} = 60,\),\( \sigma _1^2 = 10, \sigma _2^2 = 12, and \sigma _3^2 = 14\) calculate \(P\left( {{X_1} + {X_2} + {X_3} \le 160} \right)\)and also \(P\left( {{X_1} + {X_2} \ge 2{X_3}} \right).\)

Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of \({\rm{\bar X}}\) when the population distribution is Weibull with \({\rm{\alpha = 2}}\) and\({\rm{\beta = 5}}\), as in Example\({\rm{5}}{\rm{.20}}\).[A1] Consider the four sample sizes, and\({\rm{30}}\), and in each case use \({\rm{1000}}\) replications. For which of these sample sizes does the \({\rm{\bar X}}\) sampling distribution appear to be approximately normal?

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