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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

There are \({\rm{40}}\) students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of \({\rm{6}}\)min and a standard deviation of \({\rm{6}}\)min.

a. If grading times are independent and the instructor begins grading at \({\rm{6:50}}\) p.m. and grades continuously, what is the (approximate) probability that he is through grading before the \({\rm{11:00}}\) p.m. TV news begins?

b. If the sports report begins at \({\rm{11:10,}}\) what is the probability that he misses part of the report if he waits until grading is done before turning on the TV?

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 pdf of pressures for right and left front tires.

a. Determine the conditional pdf of \({\rm{Y}}\) given that \({\rm{X = x}}\) and the conditional pdf of \({\rm{X}}\) given that \({\rm{Y = y}}\).

b. If the pressure in the right tire is found to be \({\rm{22}}\) psi, what is the probability that the left tire has a pressure of at least \({\rm{25}}\) psi? Compare this to.

c. If the pressure in the right tire is found to be \({\rm{22}}\) psi, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire?

A particular brand of dishwasher soap is sold in three sizes: \({\rm{25oz,40oz}}\), and\({\rm{65oz}}\). Twenty percent of all purchasers select a\({\rm{25 - 0z}}\)box,\({\rm{50\% }}\)select a\({\rm{40 - 0z}}\)box, and the remaining\({\rm{30\% }}\)choose a\({\rm{65}}\)-oz box. Let\({{\rm{X}}_{\rm{1}}}\)and\({{\rm{X}}_{\rm{2}}}\)denote the package sizes selected by two independently selected purchasers.

a. Determine the sampling distribution of\({\rm{\bar X}}\), calculate\({\rm{E(\bar X)}}\), and compare to\({\rm{\mu }}\).

b. Determine the sampling distribution of the sample variance\({{\rm{S}}^{\rm{2}}}\), calculate\({\rm{E}}\left( {{{\rm{S}}^{\rm{2}}}} \right)\), and compare to\({{\rm{\sigma }}^{\rm{2}}}\).

Return to the situation described in Exercise \({\rm{3}}\).

a. Determine the marginal pmf of \({{\rm{X}}_{\rm{1}}}\), and then calculate the expected number of customers in line at the express checkout.

b. Determine the marginal pmf of \({{\rm{X}}_{\rm{2}}}\).

c. By inspection of the probabilities \({\rm{P(}}{{\rm{X}}_{\rm{1}}}{\rm{ = 4),P(}}{{\rm{X}}_{\rm{2}}}{\rm{ = 0),}}\) and \({\rm{P(}}{{\rm{X}}_{\rm{1}}}{\rm{ = 4,}}{{\rm{X}}_{\rm{2}}}{\rm{ = 0),}}\) are \({{\rm{X}}_{\rm{1}}}\) and \({{\rm{X}}_{\rm{2}}}\) independent random variables? Explain
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