# statistics work sheet 1

__STAT 200 Week 4 Homework Problems__

**6.1.2**

- The commuter trains on the Red Line for the Regional Transit Authority (RTA) in Cleveland, OH, have a waiting time during peak rush hour periods of eight minutes (“2012 annual report,” 2012).

a.)State the random variable.

b.)Find the height of this uniform distribution.

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Get Startedc.)Find the probability of waiting between four and five minutes.

d.)Find the probability of waiting between three and eight minutes.

e.)Find the probability of waiting five minutes exactly.

** **

**6.3.2**

Find the *z*-score corresponding to the given area.Remember, *z* is distributed as the standard normal distribution with mean of and standard deviation .

a.)The area to the left of *z* is 15%.

b.)The area to the right of *z* is 65%.

c.)The area to the left of *z* is 10%.

d.)The area to the right of *z* is 5%.

e.)The area between and *z* is 95%.(Hint draw a picture and figure out the area to the left of the .)

f.)The area between and *z* is 99%.

**6.3.4**

According to the WHO MONICA Project the mean blood pressure for people in China is 128 mmHg with a standard deviation of 23 mmHg (Kuulasmaa, Hense & Tolonen, 1998).Assume that blood pressure is normally distributed.

- State the random variable.
- Find the probability that a person in China has blood pressure of 135 mmHg or more.
- Find the probability that a person in China has blood pressure of 141 mmHg or less.
- Find the probability that a person in China has blood pressure between 120 and 125 mmHg.
- Is it unusual for a person in China to have a blood pressure of 135 mmHg?Why or why not?
- What blood pressure do 90% of all people in China have less than?

**6.3.8**

A dishwasher has a mean life of 12 years with an estimated standard deviation of 1.25 years (“Appliance life expectancy,” 2013).Assume the life of a dishwasher is normally distributed.

- State the random variable.
- Find the probability that a dishwasher will last more than 15 years.
- Find the probability that a dishwasher will last less than 6 years.
- Find the probability that a dishwasher will last between 8 and 10 years.
- If you found a dishwasher that lasted less than 6 years, would you think that you have a problem with the manufacturing process?Why or why not?
- A manufacturer of dishwashers only wants to replace free of charge 5% of all dishwashers.How long should the manufacturer make the warranty period?

**6.3.10**

The mean yearly rainfall in Sydney, Australia, is about 137 mm and the standard deviation is about 69 mm (“Annual maximums of,” 2013).Assume rainfall is normally distributed.

- State the random variable.
- Find the probability that the yearly rainfall is less than 100 mm.
- Find the probability that the yearly rainfall is more than 240 mm.
- Find the probability that the yearly rainfall is between 140 and 250 mm.
- If a year has a rainfall less than 100mm, does that mean it is an unusually dry year?Why or why not?
- What rainfall amount are 90% of all yearly rainfalls more than?

**6.4.4**

Annual rainfalls for Sydney, Australia are given in table #6.4.6. (“Annual maximums of,” 2013).Can you assume rainfall is normally distributed?

**Table #6.4.6: Annual Rainfall in Sydney, Australia**

146.8 |
383 |
90.9 |
178.1 |
267.5 |
95.5 |
156.5 |
180 |

90.9 |
139.7 |
200.2 |
171.7 |
187.2 |
184.9 |
70.1 |
58 |

84.1 |
55.6 |
133.1 |
271.8 |
135.9 |
71.9 |
99.4 |
110.6 |

47.5 |
97.8 |
122.7 |
58.4 |
154.4 |
173.7 |
118.8 |
88 |

84.6 |
171.5 |
254.3 |
185.9 |
137.2 |
138.9 |
96.2 |
85 |

45.2 |
74.7 |
264.9 |
113.8 |
133.4 |
68.1 |
156.4 |

**6.5.2**

A random variable is normally distributed.It has a mean of 245 and a standard deviation of 21.

- If you take a sample of size 10, can you say what the shape of the distribution for the sample mean is?Why?
- For a sample of size 10, state the mean of the sample mean and the standard deviation of the sample mean.
- For a sample of size 10, find the probability that the sample mean is more than 241.
- If you take a sample of size 35, can you say what the shape of the distribution of the sample mean is?Why?
- For a sample of size 35, state the mean of the sample mean and the standard deviation of the sample mean.
- For a sample of size 35, find the probability that the sample mean is more than 241.
- Compare your answers in part c and f.Why is one smaller than the other?

**6.5.4**

According to the WHO MONICA Project the mean blood pressure for people in China is 128 mmHg with a standard deviation of 23 mmHg (Kuulasmaa, Hense & Tolonen, 1998).Blood pressure is normally distributed.

- State the random variable.
- Suppose a sample of size 15 is taken.State the shape of the distribution of the sample mean.
- Suppose a sample of size 15 is taken.State the mean of the sample mean.
- Suppose a sample of size 15 is taken.State the standard deviation of the sample mean.
- Suppose a sample of size 15 is taken.Find the probability that the sample mean blood pressure is more than 135 mmHg.
- Would it be unusual to find a sample mean of 15 people in China of more than 135 mmHg?Why or why not?
- If you did find a sample mean for 15 people in China to be more than 135 mmHg, what might you conclude?

**6.5.6**

The mean cholesterol levels of women age 45-59 in Ghana, Nigeria, and Seychelles is 5.1 mmol/l and the standard deviation is 1.0 mmol/l (Lawes, Hoorn, Law & Rodgers, 2004).Assume that cholesterol levels are normally distributed.

- State the random variable.
- Find the probability that a woman age 45-59 in Ghana has a cholesterol level above 6.2 mmol/l (considered a high level).
- Suppose doctors decide to test the womanâ€™s cholesterol level again and average the two values.Find the probability that this womanâ€™s mean cholesterol level for the two tests is above 6.2 mmol/l.
- Suppose doctors being very conservative decide to test the womanâ€™s cholesterol level a third time and average the three values.Find the probability that this womanâ€™s mean cholesterol level for the three tests is above 6.2 mmol/l.
- If the sample mean cholesterol level for this woman after three tests is above 6.2 mmol/l, what could you conclude?

**6.5.8**

A dishwasher has a mean life of 12 years with an estimated standard deviation of 1.25 years (“Appliance life expectancy,” 2013).The life of a dishwasher is normally distributed.Suppose you are a manufacturer and you take a sample of 10 dishwashers that you made.

- State the random variable.
- Find the mean of the sample mean.
- Find the standard deviation of the sample mean.
- What is the shape of the sampling distribution of the sample mean?Why?
- Find the probability that the sample mean of the dishwashers is less than 6 years.
- If you found the sample mean life of the 10 dishwashers to be less than 6 years, would you think that you have a problem with the manufacturing process?Why or why not?

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