EXERCISES Section 5.4 (46-57)

46

Young’s 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 $70\mathrm{GPa}$ and $1.6\mathrm{GPa}$ , respectively (values given in the article “Influence of Material Properties Variability on Springback and Thinning in Sheet

Stamping Processes: A Stochastic Analysis” (Intl. J. of Advanced Manuf. Tech., 2010: 117-134)).

a. If $\bar{X}$ is the sample mean Young’s modulus for a random sample of $n=16$ sheets, where is the sampling distribution of $\bar{X}$ centered, and what is the standard deviation of the $\bar{X}$ distribution?

b. Answer the questions posed in part (a) for a sample size of $n=64$ sheets.

c. For which of the two random samples, the one of part (a) or the one of part (b), is $\bar{X}$ more likely to be within $1\mathrm{GPa}$ of $70\mathrm{GPa}$ ? Explain your reasoning.

47

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

a. Calculate $P\left(69\le \bar{X}\le 71\right)$ when $n=16$ .

b. How likely is it that the sample mean diameter exceeds 71 when $n=25$ ?

48

The National Health Statistics Reports dated Oct. 22, 2008, stated that for a sample size of 27718-year-old American males, the sample mean waist circumference was $86.3\mathrm{cm}$ . A somewhat complicated method was used to estimate various population percentiles, resulting in the following values: $\begin{array}{lllllll}{5}^{\text{th }}& {10}^{\text{th }}& {25}^{\text{th }}& {50}^{\text{th }}& {75}^{\text{th }}& {90}^{\text{th }}& {95}^{\text{th }}\end{array}$ $\begin{array}{lllllll}69.6& 70.9& 75.2& 81.3& 95.4& 107.1& 116.4\end{array}$

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 $85\mathrm{cm}$ and that the population standard deviation is $15\mathrm{cm}$ . How likely is it that a random sample of 277 individuals will result in a sample mean waist size of at least $86.3\mathrm{cm}$ ?

c. Referring back to (b), suppose now that the population mean waist size in $82\mathrm{cm}$ . Now what is the (approximate) probability that the sample mean will be at least $86.3\mathrm{cm}$ ? In light of this calculation, do you think that $82\mathrm{cm}$ is a reasonable value for $\mu$ ?

49

There are 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 $6\min$ and a standard deviation of $6\min$ .

a. If grading times are independent and the instructor begins grading at $6:50$ P.M. and grades continuously, what is the (approximate) probability that he is through grading before the $11:00$ P.M. TV news begins?

b. If the sports report begins at 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?

50

Let $X$ denote the courtship time for a randomly selected female-male pair of mating scorpion flies (time from the beginning of interaction until mating). Suppose the mean value of $X$ is $120\min$ and the standard deviation of $X$ is 110 min (suggested by data in the article “Should I Stay or Should I Go? Condition- and Status-Dependent Courtship Decisions in the Scorpion Fly Panorpa Cognate” (Animal Behavior, 2009: 491-497)).

a. Is it plausible that $X$ is normally distributed?

b. For a random sample of 50 such pairs, what is the (approximate) probability that the sample mean courtship time is between $100\min$ and $125\min$ ?

c. For a random sample of 50 such pairs, what is the (approximate) probability that the total courtship time exceeds $150\mathrm{hr}$ ?

d. Could the probability requested in (b) be calculated from the given information if the sample size were 15 rather than 50 ? Explain.

51

The time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal distribution with mean value $10\min$ and standard deviation 2 min. If five individuals fill out a form on one day and six on another, what is the probability that the sample average amount of time taken on each day is at most $11\min$ ?

52

The lifetime of a certain type of battery is normally distributed with mean value 10 hours and standard deviation 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 $5\%$ of all packages?

53

Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.2 .

a. If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 9 pins is at least 51 ?

b. Without assuming population normality, what is the (approximate) probability that the sample mean hardness for a random sample of 40 pins is at least 51?

54

Suppose the sediment density $\left(\mathrm{g}/\mathrm{cm}\right)$ of a randomly selected specimen from a certain region is normally distributed with mean 2.65 and standard deviation .85 (suggested in “Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants,” Water Research, 1984: 1169-1174).

a. If a random sample of 25 specimens is selected, what is the probability that the sample average sediment density is at most 3.00 ? Between 2.65 and 3.00 ?

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

55

The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter $\mu =50$ .

a. Calculate the approximate probability that between 35 and 70 tickets are given out on a particular day.

b. Calculate the approximate probability that the total number of tickets given out during a 5-day week is between 225 and 275.

c. Use software to obtain the exact probabilities in (a) and (b) and compare to their approximations.

56

A binary communication channel transmits a sequence of “bits” (0s and 1s). Suppose that for any particular bit transmitted, there is a $10\%$ chance of a transmission error (a 0 becoming a 1 or a 1 becoming a 0 ). Assume that bit errors occur independently of one another.

a. Consider transmitting 1000 bits. What is the approximate probability that at most 125 transmission errors occur?

b. Suppose the same 1000-bit message is sent two different times independently of one another. What is the approximate probability that the number of errors in the first transmission is within 50 of the number of errors in the second?

57

Suppose the distribution of the time $X$ (in hours) spent by students at a certain university on a particular project is gamma with parameters $\alpha =50$ and $\beta =2$ . Because $\alpha$ is large, it can be shown that $X$ has approximately a normal distribution. Use this fact to compute the approximate probability that a randomly selected student spends at most 125 hours on the project.