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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

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 \(P\left( {{X_1} + {X_2} \ge 2{X_3}} \right).\)

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.

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?

Refer to Exercise \({\rm{46}}\). Suppose the distribution is normal (the cited article makes that assumption and even includes the corresponding normal density curve).

a. Calculate \({\rm{P}}\)(\(69\text{拢}\bar{X}\text{拢}71\)) when \({\rm{n = 16}}\).

b. How likely is it that the sample mean diameter exceeds \({\rm{71}}\) when \({\rm{n = 25}}\)?

Let \({\rm{X}}\) denote the number of Canon SLR cameras sold during a particular week by a certain store. The pmf of \({\rm{X}}\) is

Sixty percent of all customers who purchase these cameras also buy an extended warranty. Let \({\rm{Y}}\) denote the number of purchasers during this week who buy an extended warranty.

a. What is\({\rm{P(X = 4,Y = 2)}}\)? (Hint: This probability equals\({\rm{P(Y = 2}}\mid {\rm{X = 4) \times P(X = 4)}}\); now think of the four purchases as four trials of a binomial experiment, with success on a trial corresponding to buying an extended warranty.)

b. Calculate\({\rm{P(X = Y)}}\).

c. Determine the joint pmf of \({\rm{X}}\) and \({\rm{Y}}\)then the marginal pmf of\({\rm{Y}}\).
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