7.10 Central limit theorem: homework

$X$ ~ $N(\text{60},9)$ . Suppose that you form random samples of 25 from this distribution. Let $\overline{X}$ be the random variable of averages. Let $\mathrm{\Sigma X}$ be the random variable of sums. For c - f , sketch the graph, shade the region, label and scale the horizontal axis for $\overline{X}$ , and find the probability.

• a Sketch the distributions of $X$ and $\overline{X}$ on the same graph.
• b $\overline{X}$ ~
• c
• e
• f
• g $\mathrm{\Sigma x}$ ~
• h Find the minimum value for the upper quartile for the sum.
• i

Determine which of the following are true and which are false. Then, in complete sentences, justify your answers.

• a When the sample size is large, the mean of $\overline{X}$ is approximately equal to the mean of $X$ .
• b When the sample size is large, $\overline{X}$ is approximately normally distributed.
• c When the sample size is large, the standard deviation of $\overline{X}$ is approximately the same as the standard deviation of $X$ .

The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about 36 and a standard deviation of about 10. Suppose that 16 individuals are randomly chosen.

Let $\overline{X}=$ average percent of fat calories.

• a $\overline{X}\text{~}$ ______ ( ______ , ______ )
• b For the group of 16, find the probability that the average percent of fat calories consumed is more than 5. Graph the situation and shade in the area to be determined.
• c Find the first quartile for the average percent of fat calories.

Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.

• a If $\overline{X}=$ average distance in feet for 49 fly balls, then $\overline{X}\text{~}$ ______ ( ______ , ______ )
• b What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for $\overline{X}$ . Shade the region corresponding to the probability. Find the probability.
• c Find the 80th percentile of the distribution of the average of 49 fly balls.

Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from 2 to 6 pounds. We randomly survey 64 homes with children.

• a In words, $X=$
• b $X\text{~}$
• c ${\mu }_X=$
• d ${\sigma }_X=$
• e In words, $\mathrm{\Sigma X}=$
• f $\mathrm{\Sigma X}\text{~}$
• g Find the probability that the total weight of open boxes is less than 250 pounds.
• h Find the 35th percentile for the total weight of open boxes of cereal.

Suppose that the duration of a particular type of criminal trial is known to have a mean of 21 days and a standard deviation of 7 days. We randomly sample 9 trials.

• a In words, $\mathrm{\Sigma X}=$
• b $\mathrm{\Sigma X}\text{~}$
• c Find the probability that the total length of the 9 trials is at least 225 days.
• d 90 percent of the total of 9 of these types of trials will last at least how long?

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.