8.1 Homework  (Page 3/5)

A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length of time they last is approximately normal. Researchers in a hospital used the drug on a random sample of 9 patients. The effective period of the tranquilizer for each patient (in hours) was as follows: 2.7; 2.8; 3.0; 2.3; 2.3; 2.2; 2.8; 2.1; and 2.4 .

• a
  • i $\overline{x}=\text{\_\_\_\_\_\_\_\_}$
  • ii $s_x=\text{\_\_\_\_\_\_\_\_}$
  • iii $n=\text{\_\_\_\_\_\_\_\_}$
  • iv $n-1=\text{\_\_\_\_\_\_\_\_}$
• b Define the Random Variable $X$ , in words.
• c Define the Random Variable $\overline{X}$ , in words.
• d Which distribution should you use for this problem? Explain your choice.
• e Construct a 95% confidence interval for the population mean length of time.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the error bound.
• f What does it mean to be “95% confident” in this problem?

Suppose that 14 children were surveyed to determine how long they had to use training wheels. It was revealed that they used them an average of 6 months with a sample standard deviation of 3 months. Assume that the underlying population distribution is normal.

• a
  • i $\overline{x}=\text{\_\_\_\_\_\_\_\_}$
  • ii $s_x=\text{\_\_\_\_\_\_\_\_}$
  • iii $n=\text{\_\_\_\_\_\_\_\_}$
  • iv $n-1=\text{\_\_\_\_\_\_\_\_}$
• b Define the Random Variable $X$ , in words.
• c Define the Random Variable $\overline{X}$ , in words.
• d Which distribution should you use for this problem? Explain your choice.
• e Construct a 99% confidence interval for the population mean length of time using training wheels.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the error bound.
• f Why would the error bound change if the confidence level was lowered to 90%?

Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car.

• a When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03?
• b If it was later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?

Suppose that the insurance companies did do a survey. They randomly surveyed 400 drivers and found that 320 claimed to always buckle up. We are interested in the population proportion of drivers who claim to always buckle up.

• a
  • i $x=\text{\_\_\_\_\_\_\_\_}$
  • ii $n=\text{\_\_\_\_\_\_\_\_}$
  • iii $p\text{'}=\text{\_\_\_\_\_\_\_\_}$
• b Define the Random Variables $X$ and $P\text{'}$ , in words.
• c Which distribution should you use for this problem? Explain your choice.
• d Construct a 95% confidence interval for the population proportion that claim to always buckle up.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the error bound.
• e If this survey were done by telephone, list 3 difficulties the companies might have in obtaining random results.

Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. To accomplish this, the records of 225 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats.

• a
  • i $\overline{x}=\text{\_\_\_\_\_\_\_\_}$
  • ii $s_x=\text{\_\_\_\_\_\_\_\_}$
  • iii $n=\text{\_\_\_\_\_\_\_\_}$
  • iv $n-1=\text{\_\_\_\_\_\_\_\_}$
• b Define the Random Variables $X$ and $\overline{X}$ , in words.
• c Which distribution should you use for this problem? Explain your choice.
• d Construct a 92% confidence interval for the population mean number of unoccupied seats per flight.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the error bound.