8.13 Confidence intervals: homework  (Page 2/5)

Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for service. The committee randomly surveyed 81 people. The sample mean was 8 hours with a sample standard deviation of 4 hours.

• 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 95% confidence interval for the population mean time wasted.
  • a State the confidence interval.
  • b Sketch the graph.
  • c Calculate the margin of error.
• e Explain in a complete sentence what the confidence interval means.

Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. It randomly surveys 100 people. The sample mean is 23.6 hours. There is a known standard deviation of 7.0 hours. The population distribution is assumed to be normal.

• a
  • i $\overline{x}=$ $\text{\_\_\_\_\_\_\_\_}$
  • ii $\sigma =$ $\text{\_\_\_\_\_\_\_\_}$
  • iii $s_x=$ $\text{\_\_\_\_\_\_\_\_}$
  • iv $n=$ $\text{\_\_\_\_\_\_\_\_}$
  • v $n-1=$ $\text{\_\_\_\_\_\_\_\_}$
• 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 90% confidence interval for the population mean time to complete the tax forms.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the margin of error.
• e If the firm wished to increase its level of confidence and keep the margin of error the same by taking another survey, what changes should it make?
• f If the firm did another survey, kept the margin of error the same, and only surveyed 49 people, what would happen to the level of confidence? Why?
• g Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within 1 hour. How would the number of people the firm surveys change? Why?

A sample of 16 small bags of the same brand of candies was selected. Assume that the population distribution of bag weights is normal. The weight of each bag was then recorded. The mean weight was 2 ounces with a standard deviation of 0.12 ounces. The population standard deviation is known to be 0.1 ounce.

• a
  • i $\overline{x}=$ $\text{\_\_\_\_\_\_\_\_}$
  • ii $\sigma =$ $\text{\_\_\_\_\_\_\_\_}$
  • iii $s_x=$ $\text{\_\_\_\_\_\_\_\_}$
  • iv $n=$ $\text{\_\_\_\_\_\_\_\_}$
  • v $n-1=$ $\text{\_\_\_\_\_\_\_\_}$
• 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 90% confidence interval for the population mean weight of the candies.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the margin of error.
• f Construct a 98% confidence interval for the population mean weight of the candies.
  • i State the confidence interval.
  • ii Sketch the graph.
  • iii Calculate the margin of error.
• g In complete sentences, explain why the confidence interval in (f) is larger than the confidence interval in (e).
• h In complete sentences, give an interpretation of what the interval in (f) means.