Module 7: The Central Limit Theorem

# Section Exercises

Barbara Illowsky & OpenStax et al.

## The Central Limit Theorem for Sample Means (Averages)

*Use the following information to answer the next six exercises:*Yoonie is a personnel manager in a large corporation. Each month she must review 16 of the employees. From past experience, she has found that the reviews take her approximately four hours each to do with a population standard deviation of 1.2 hours. Let

*Χ*be the random variable representing the time it takes her to complete one review. Assume

*Χ*is normally distributed. Let[latex]displaystyleoverline{{X}}[/latex] be the random variable representing the mean time to complete the 16 reviews. Assume that the 16 reviews represent a random set of reviews.

*X*~ _____(_____,_____)

**one**review will take Yoonie from 3.5 to 4.25 hours. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.

*P*(________ <

*x*< ________) = _______

**mean**of a month’s reviews will take Yoonie from 3.5 to 4.25 hrs. Sketch the graph, labeling and scaling the horizontal axis. Shade the region corresponding to the probability.

*5. P*(________________) = _______

^{th}percentile for the mean time to complete one month’s reviews. Sketch the graph.The 95

^{th}Percentile =____________

- In words,
*Χ*= ____________ *Χ*~ _____(_____,_____)- In words,[latex]displaystyleoverline{{X}}[/latex]= ____________
- [latex]displaystyleoverline{{X}}[/latex]~ ______ (______, ______)
- Find the probability that an individual had between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.
- Find the probability that the average of the 25 students was between $0.80 and $1.00. Graph the situation, and shade in the area to be determined.
- Explain why there is a difference in part e and part f.

- If[latex]displaystyleoverline{{X}}[/latex] = average distance in feet for 49 fly balls, then[latex]displaystyleoverline{{X}}[/latex] ~ _______(_______,_______)
- What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for[latex]displaystyleoverline{{X}}[/latex]. Shade the region corresponding to the probability. Find the probability.
- Find the 80
^{th}percentile of the distribution of the average of 49 fly balls.

- In words,
*Χ*= _____________ - In words,[latex]displaystyleoverline{{X}}[/latex]= _____________
- [latex]displaystyleoverline{{X}}[/latex] ~ _____(_____,_____)
- 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.
- Would you be surprised if one taxpayer finished his or her Form 1040 in more than 12 hours? In a complete sentence, explain why.

- [latex]displaystyleoverline{{X}}[/latex] ~ _____(_____,_____)
- Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons.
- Find the 80
^{th}percentile for the average of these 49 marathons. - Find the median of the average running times.

- In words,
*Χ*= _________ *Χ*~ _____________- In words,[latex]displaystyleoverline{{X}}[/latex] = _____________
- [latex]displaystyleoverline{{X}}[/latex] ~ _____(_____,_____)
- Find the first quartile for the average song length.
- The IQR(interquartile range) for the average song length is from _______–_______.

- In words,
*Χ*= _____________ - In words,[latex]displaystyleoverline{{X}}[/latex] = _____________
- [latex]displaystyleoverline{{X}}[/latex]~ _____(_____,_____)
- The IQR for[latex]displaystyleoverline{{X}}[/latex] is from _______ acres to _______ acres.

- When the sample size is large, the mean of[latex]displaystyleoverline{{X}}[/latex] is approximately equal to the mean of
*Χ*. - When the sample size is large,[latex]displaystyleoverline{{X}}[/latex] is approximately normally distributed.
- When the sample size is large, the standard deviation of[latex]displaystyleoverline{{X}}[/latex] is approximately the same as the standard deviation of
*Χ*.

- [latex]displaystyleoverline{{X}}[/latex] ~ ______(______, ______)
- For the group of 16, find the probability that the average percent of fat calories consumed is more than five. Graph the situation and shade in the area to be determined.
- Find the first quartile for the average percent of fat calories.

- In words,
*Χ*= _____________ - In words,[latex]displaystyleoverline{{X}}[/latex] = _____________
- [latex]displaystyleoverline{{X}}[/latex] ~ _____(_____,_____)
- How is it possible for the standard deviation to be greater than the average?
- Why is it more likely that the average of the 1,000 residents will be from $2,000 to $2,100 than from $2,100 to $2,200?

- The mean, median, and mode are equal.
- The area under the curve is one.
- The curve never touches the
*x*-axis. - The curve is skewed to the right.

*N*(4.59, 0.10)

*N*(4.59, [latex]frac{{0.10}}{{sqrt{16}}}[/latex])>

*N*(4.59, [latex]frac{{16}}{{0.10}}[/latex])

*N*(4.59, [latex]frac{{sqrt{16}}}{{0.10}}[/latex])

## The Central Limit Theorem for Sums

*Use the following information to answer the next four exercises:*An unknown distribution has a mean of 80 and a standard deviation of 12. A sample size of 95 is drawn randomly from the population.

*Use the following information to answer the next five exercises:* The distribution of results from a cholesterol test has a mean of 180 and a standard deviation of 20. A sample size of 40 is drawn randomly.

*Use the following information to answer the next six exercises:* A researcher measures the amount of sugar in several cans of the same soda. The mean is 39.01 with a standard deviation of 0.5. The researcher randomly selects a sample of 100.

*z*–score of –2.5.

*z*–score of 0.5.

*z*-scores –2 and 1.

*Use the following information to answer the next four exercise:* An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let *X* = the object of interest.

*ΣX*?

*ΣX*?

*P*(

*Σx*= 290)?

*P*(

*Σx*> 290)?

37. True or False: only the sums of normal distributions are also normal distributions.

*X*= one object from this distribution. What is the sample size if the standard deviation of

*ΣX*is 42?

41. An unknown distribution has a mean of 19 and a standard deviation of 20. Let *X* = the object of interest. What is the sample size if the mean of *ΣX* is 15,200?

*Use the following information to answer the next three exercises.* A market researcher analyzes how many electronics devices customers buy in a single purchase. The distribution has a mean of three with a standard deviation of 0.7. She samples 400 customers.

*z*-score for

*Σx*= 840?

*z*-score for

*Σx*= 1,186?

*P*(

*Σx*< 1,186)?

*Use the following information to answer the next three exercises:* An unkwon distribution has a mean of 100, a standard deviation of 100, and a sample size of 100. Let *X* = one object of interest.

*ΣX*?

*ΣX*?

*P*(

*Σx*> 9,000)?

- The mean, median and mode are equal.
- The area under the curve is one.
- The curve never touches the
*x*-axis. - The curve is skewed to the right.

- In words,
*ΣX*= ______________ *ΣX*~ _____(_____,_____)- Find the probability that the total length of the nine trials is at least 225 days.
- Ninety percent of the total of nine of these types of trials will last at least how long?

50.Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from two to six pounds with a mean of four pounds and standard deviation of 1.1547. We randomly survey 64 homes with children.

- In words,
*X*= _____________ - The distribution is _______.
- In words,
*ΣX*= _______________ *ΣX*~ _____(_____,_____)- Find the probability that the total weight of open boxes is less than 250 pounds.
- Find the 35
^{th}percentile for the total weight of open boxes of cereal.

- In words,
*X*= ______________ *X*~ _____(_____,_____)- In words,
*ΣX*= _____________ *ΣX*~ _____(_____,_____)- Find the probability that the teachers earn a total of over $400,000.
- Find the 90
^{th}percentile for an individual teacher’s salary. - Find the 90
^{th}percentile for the sum of ten teachers’ salary. - If we surveyed 70 teachers instead of ten, graphically, how would that change the distribution in part d?
- If each of the 70 teachers received a $3,000 raise, graphically, how would that change the distribution in part b?

## Using the Central Limit Theorem

*Use the following information to answer the next 8 exercises:* A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100 weights is taken.

52. What is the distribution for the weights of one 25-pound lifting weight? What is the mean and standard deivation?

53. What is the distribution for the mean weight of 100 25-pound lifting weights?

54. Find the probability that the mean actual weight for the 100 weights is less than 24.9.

55. Find the probability that the mean actual weight for the 100 weights is greater than 25.2.

^{th}percentile for the mean weight for the 100 weights.

*P*(

*Σx*< 2,450).

^{th}percentile for the total weight of the 100 weights.

*Use the following information to answer the next 9 exercises:* The length of time a particular smartphone’s battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.

60. What is the standard deviation?

61. What is the parameter *m*?

^{th}percentile for the total length of time 64 batteries last.

*IQR*for the mean amount of time 64 batteries last.

*Use the following information to answer the next 4 exercises:* A uniform distribution has a minimum of six and a maximum of ten. A sample of 50 is taken.

*P*(

*Σx*> 420).

^{th}percentile for the sums.

^{th}percentile for the sums.

^{th}percentile for the sums.

- In words,
*Χ*= _______ *Χ*~ _____(_____,_____)- In words,[latex]displaystyleoverline{{X}}[/latex]= ____________
- [latex]displaystyleoverline{{X}}[/latex] ~ _____(_____,_____)
- Before doing any calculations, which do you think will be higher? Explain why.
- The probability that an individual attention span is less than ten minutes.
- The probability that the average attention span for the 60 children is less than ten minutes?

- Calculate the probabilities in part e.
- Explain why the distribution for[latex]displaystyleoverline{{X}}[/latex] is not exponential.

- In words,
*Χ*= ______________ -
- [latex]displaystyleoverline{{x}}[/latex] = _____
*s*= ______{x}*n*= _____

- Construct a histogram of the distribution of the averages. Start at
*x*= –0.0005. Use bar widths of ten. - In words, describe the distribution of stock prices.
- Randomly average five stock prices together. (Use a random number generator.) Continue averaging five pieces together until you have ten averages. List those ten averages.
- Use the ten averages from part e to calculate the following.
- [latex]displaystyleoverline{{x}}[/latex] = _____
*s*= ______{x}

- Construct a histogram of the distribution of the averages. Start at
*x*= -0.0005. Use bar widths of ten. - Does this histogram look like the graph in part c?
- In one or two complete sentences, explain why the graphs either look the same or look different?
- Based upon the theory of the
**central limit theorem**,[latex]displaystyleoverline{{X}}[/latex] ~ _____(_____,____)

*Use the following information to answer the next three exercises:* Richard’s Furniture Company delivers furniture from 10 A.M. to 2 P.M. continuously and uniformly. We are interested in how long (in hours) past the 10 A.M. start time that individuals wait for their delivery.

*75. Χ*~ _____(_____,_____)

*U*(0,4)*U*(10,2)*Eχp*(2)*N*(2,1)

- one hour.
- two hours.
- two and a half hours.
- four hours.

**more**hours is:

- [latex]frac{1}{4}[/latex]
- [latex]frac{1}{2}[/latex]
- [latex]frac{3}{4}[/latex]
- [latex]frac{3}{8}[/latex]

*Use the following information to answer the next two exercises:* The time to wait for a particular rural bus is distributed uniformly from zero to 75 minutes. One hundred riders are randomly sampled to learn how long they waited.

^{th}percentile sample average wait time (in minutes) for a sample of 100 riders is:

- 315.0
- 40.3
- 38.5
- 65.2

- yes
- no
- There is not enough information.

*Use the following to answer the next two exercises:* The cost of unleaded gasoline in the Bay Area once followed an unknown distribution with a mean of $4.59 and a standard deviation of $0.10. Sixteen gas stations from the Bay Area are randomly chosen. We are interested in the average cost of gasoline for the 16 gas stations.

- almost zero
- 0.1587
- 0.0943
- unknown

- 0.6554
- 0.3446
- 0.0142
- 0.9858
- 0

^{th}grade (K – 12) school district, 53 percent of the population favor a charter school for grades K through five. A simple random sample of 300 is surveyed. Calculate following using the normal approximation to the binomial distribution.

- Find the probability that less than 100 favor a charter school for grades K through 5.
- Find the probability that 170 or more favor a charter school for grades K through 5.
- Find the probability that no more than 140 favor a charter school for grades K through 5.
- Find the probability that there are fewer than 130 that favor a charter school for grades K through 5.
- Find the probability that exactly 150 favor a charter school for grades K through 5.

83. If you have access to an appropriate calculator or computer software, try calculating these probabilities using the technology.

- Find the probability that Janice is the driver at most 20 days.
- Find the probability that Roberta is the driver more than 16 days.
- Find the probability that Barbara drives exactly 24 of those 96 days.

*85. X*~

*N*(60, 9). Suppose that you form random samples of 25 from this distribution. Let[latex]displaystyleoverline{{X}}[/latex] be the random variable of averages. Let

*ΣX*be the random variable of sums. For parts c through f, sketch the graph, shade the region, label and scale the horizontal axis for[latex]displaystyleoverline{{X}}[/latex], and find the probability.

- Sketch the distributions of
*X*and[latex]displaystyleoverline{{X}}[/latex]on the same graph. - [latex]displaystyleoverline{{X}}[/latex] ~ _____(_____,_____)
*P*([latex]displaystyleoverline{{x}}[/latex]< 60) = _____- Find the 30
^{th}percentile for the mean. *P*(56 <[latex]displaystyleoverline{{x}}[/latex]< 62) = _____*P*(18 <[latex]displaystyleoverline{{x}}[/latex] < 58) = _____*Σx*~ _____(_____,_____)- Find the minimum value for the upper quartile for the sum.
*P*(1,400 <*Σx*< 1,550) = _____

- In words,
*X*= _____________ *X*~ _____(_____,_____)*μ*= ______{x}*σ*= ______{x}- In words,[latex]displaystyleoverline{{X}}[/latex]= ______________
- [latex]displaystyleoverline{{X}}[/latex] ~ _____(_____,_____)
- In words,
*ΣX*= _____________ *ΣX*~ _____(_____,_____)- Without doing any calculations, do you think that it’s likely that the professor will need to read a total of more than 1,050 pages? Why?
- Calculate the probability that the professor will need to read a total of more than 1,050 pages.
- Why is it so unlikely that the average length of the papers will be less than 12 pages?

- Find the 90
^{th}percentile for an individual teacher’s salary. - Find the 90
^{th}percentile for the average teacher’s salary.

- In words,
*X*= _____________ - In words,[latex]displaystyleoverline{{X}}[/latex] = ___________________
- [latex]displaystyleoverline{{X}}[/latex] ~ _____(_____,_____)
- In words,
*ΣX*= _______________ *ΣX*~ _____(_____,_____)- Is it likely that an individual stayed more than five days in the hospital? Why or why not?
- Is it likely that the average stay for the 80 women was more than five days? Why or why not?
- Which is more likely:
- An individual stayed more than five days.
- the average stay of 80 women was more than five days.

- If we were to sum up the women’s stays, is it likely that, collectively they spent more than a year in the hospital? Why or why not?

*For each problem, wherever possible, provide graphs and use the calculator.*

- Find the probability that 20 randomly selected men will have a sum weight greater than 3600 lbs.
- If 20 men have a sum weight greater than 3500 lbs, then their total weight exceeds the safety limits for water taxis. Based on (a), is this a safety concern? Explain.

M&M candies large candy bags have a claimed net weight of 396.9 g. The standard deviation for the weight of the individual candies is 0.017 g. The following table is from a stats experiment conducted by a statistics class.

Red | Orange | Yellow | Brown | Blue | Green |
---|---|---|---|---|---|

0.751 | 0.735 | 0.883 | 0.696 | 0.881 | 0.925 |

0.841 | 0.895 | 0.769 | 0.876 | 0.863 | 0.914 |

0.856 | 0.865 | 0.859 | 0.855 | 0.775 | 0.881 |

0.799 | 0.864 | 0.784 | 0.806 | 0.854 | 0.865 |

0.966 | 0.852 | 0.824 | 0.840 | 0.810 | 0.865 |

0.859 | 0.866 | 0.858 | 0.868 | 0.858 | 1.015 |

0.857 | 0.859 | 0.848 | 0.859 | 0.818 | 0.876 |

0.942 | 0.838 | 0.851 | 0.982 | 0.868 | 0.809 |

0.873 | 0.863 | ||||

0.809 | 0.888 | ||||

0.890 | 0.925 | ||||

0.878 | 0.793 | ||||

0.905 | 0.977 | ||||

0.850 | |||||

0.830 | |||||

0.856 | |||||

0.842 | |||||

0.778 | |||||

0.786 | |||||

0.853 | |||||

0.864 | |||||

0.873 | |||||

0.880 | |||||

0.882 | |||||

0.931 | |||||

0.887 |

91. The bag contained 465 candies and he listed weights in the table came from randomly selected candies. Count the weights.

- Find the mean sample weight and the standard deviation of the sample weights of candies in the table.
- Find the sum of the sample weights in the table and the standard deviation of the sum the of the weights.
- If 465 M&Ms are randomly selected, find the probability that their weights sum to at least 396.9.
- Is the Mars Company’s M&M labeling accurate?

0.757 | 0.723 | 0.754 | 0.737 | 0.757 | 0.741 | 0.722 | 0.741 | 0.743 | 0.742 |

0.740 | 0.758 | 0.724 | 0.739 | 0.736 | 0.735 | 0.760 | 0.750 | 0.759 | 0.754 |

0.744 | 0.758 | 0.765 | 0.756 | 0.738 | 0.742 | 0.758 | 0.757 | 0.724 | 0.757 |

0.744 | 0.738 | 0.763 | 0.756 | 0.760 | 0.768 | 0.761 | 0.742 | 0.734 | 0.754 |

0.758 | 0.735 | 0.740 | 0.743 | 0.737 | 0.737 | 0.725 | 0.761 | 0.758 | 0.756 |

The screws were randomly selected from the local home repair store.

- Find the mean diameter and standard deviation for the sample
- Find the probability that 50 randomly selected screws will be within the stated tolerance levels. Is the company’s diameter claim plausible?