7.1 Homework: clt (modified r. bloom)  (Page 2/3)

According to the Internal Revenue Service, the average length of time for an individual to complete (record keep, learn, prepare, copy, assemble and send) IRS Form 1040 is 10.53 hours (without any attached schedules). The distribution is unknown. Let us assume that the standard deviation is 2 hours. Suppose we randomly sample 36 taxpayers.

• a In words, $X=$
• b In words, $\overline{X}=$
• c $\overline{X}\text{~}$
• d Would you be surprised if the 36 taxpayers finished their Form 1040s in an average of more than 12 hours? Explain why or why not in complete sentences.
• e Would you be surprised if one taxpayer finished his Form 1040 in more than 12 hours? In a complete sentence, explain why.

Suppose that a category of world class runners are known to run a marathon (26 miles) in an average of 145 minutes with a standard deviation of 14 minutes. Consider 49 of the races.

Let $\overline{X}=$ the average of the 49 races.

• a $\overline{X}\text{~}$
• b Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons.
• c Find the 80th percentile for the average of these 49 marathons.
• d Find the median of the average running times.

The attention span of a two year-old is exponentially distributed with a mean of about 8 minutes. Suppose we randomly survey 60 two year-olds.

• a In words, $X=$
• b $X\text{~}$
• c In words, $\overline{X}=$
• d $\overline{X}\text{~}$
• e Before doing any calculations, which do you think will be higher? Explain why.
  • i the probability that an individual attention span is less than 10 minutes; or
  • ii the probability that the average attention span for the 60 children is less than 10 minutes? Why?
• f Calculate the probabilities in part (e).
• g Explain why the distribution for $\overline{X}$ is not exponential.

Note: Parts g,h,i,j of this problem have been changed from original version of textbook.

Suppose that the length of research papers is uniformly distributed from 10 to 25 pages. (Use the continuous uniform distribution - assume that the page count measures fractional pages.) We survey a random sample of 55 research papers turned in to a professor. We are interested in the average length of the research papers.

• a In words, $X=$
• b $X\text{~}$
• c ${\mu }_X=$
• d ${\sigma }_X=$
• e In words, $\overline{X}=$
• f $\overline{X}\text{~}$
• g Find the probability that an individual paper is longer than 18 pages.
• h FInd the probability that the average length of the 55 papers is more than 18 pages.
• i Find the 64th percentile for the length of individual papers.
• j Find the 64th percentile for the average length for samples of papers.
• k Why is it so unlikely that the average length of the papers will be less than 12 pages?

The length of songs in a collector’s CD collection is uniformly distributed from 2 to 3.5 minutes. Suppose we randomly pick 5 CDs from the collection. There is a total of 43 songs on the 5 CDs.

• a In words, $X=$
• b $X\text{~}$
• c In words, $\overline{X}\text{=}$
• d $\overline{X}\text{~}$
• e Find the first quartile for the average song length.
• f The IQR (interquartile range) for the average song length is from _______ to _______.