Confidence Intervals Homework

Exercise 1

Among various ethnic groups, the standard deviation of heights is known to be approximately 3 inches. We wish to construct a 95% confidence interval for the average height of male Swedes. 48 male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches.

1. 
   
   1. $\overline x$=
   2. $σ$=
      
   3. $s_x$ =
      
   4. $n - 1$ =
      
      
2. Define the Random Variables X and $\overline X$, in words.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 95% confidence interval for the population average height of male Swedes.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
5. What will happen to the level of confidence obtained if 1000 male Swedes are surveyed instead of 48? Why?

Exercise 2

A random survey of enrollment at 35 community colleges across the United States yielded the following figures (source: Microsoft Bookshelf): 6414; 1550; 2109; 9350; 21828; 4300; 5944; 5722; 2825; 2044; 5481; 5200; 5853; 2750; 10012; 6357; 27000; 9414; 7681; 3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493; 2771; 2861; 1263; 7285; 28165; 5080; 11622. Assume the underlying population is normal.

1. 
   
   1. $\overline X$=
   2. $σ$=
      
   3. $s_x$ =
      
   4. $n - 1$ =
      
2. Define the Random Variables X and $\overline X$, in words.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 95% confidence interval for the population average enrollment at community colleges in the United States.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
5. What will happen to the error bound and confidence interval if 500 community colleges were surveyed? Why?

Exercise 3

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

1. 
   
1. $\overline X$=
2. $σ$=
   
3. $s_x$ =
   
4. $n - 1$ =
   
2. Define the Random Variables X and $\overline X$, in words.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 95% confidence interval for the population average time wasted.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
5. Explain in a complete sentence what the confidence interval means.

Exercise 4

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.

1. $\overline X$=
2. $σ$=
   
3. $s_x$ =
   
4. $n - 1$ =
2. Define the Random Variable X, in words.
3. Define the Random Variable $\overline X$, in words.
4. Which distribution should you use for this problem? Explain your choice.
5. Construct a 90% confidence interval for the population average weight of the candies.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
6. Construct a 98% confidence interval for the population average weight of the candies.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
7. In complete sentences, explain why the confidence interval in (f) is larger than the confidence interval in (e).
   
8. h. In complete sentences, give an interpretation of what the interval in (f) means.

Exercise 5

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.

1. $\overline X$=
2. $σ$=
   
3. $s_x$ =
   
4. $n - 1$ =
   
2. Define the Random Variable X, in words.
3. Define the Random Variable $\overline X$, in words.
4. Which distribution should you use for this problem? Explain your choice.
5. Construct a 99% confidence interval for the population average length of time using training wheels.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
6. Why would the error bound change if the confidence level was lowered to 90%?

Exercise 6

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.

1. 
   
   1. $x$ =
   2. $n$=
   3. $p'$=
      
2. Define the Random Variables X and P', in words.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 95% confidence interval for the population proportion that claim to always buckle up.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
5. If this survey were done by telephone, list 3 difficulties the companies might have in obtaining random results.

Exercise 7

According to a recent survey of 1200 people, 61% feel that the president is doing an acceptable job. We are interested in the population proportion of people who feel the president is doing an acceptable job.

1. Define the Random Variables X and P', in words.
2. Which distribution should you use for this problem? Explain your choice.
3. Construct a 90% confidence interval for the population proportion of people who feel the president is doing an acceptable job.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.

Exercise 8

A camp director is interested in the average number of letters each child sends during his/her camp session. The population standard deviation is known to be 2.5. A survey of 20 campers is taken. The average from the sample is 7.9 with a sample standard deviation of 2.8.

1. $\overline X$=
2. $σ$=
   
3. $s_x$ =
4. $n$ =
   
5. $n - 1$ =
   
2. Define the Random Variables X and $\overline X$, in words.
3. Which distribution should you use for this problem? Explain your choice.
4. Construct a 90% confidence interval for the population average number of letters campers send home.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
5. What will happen to the error bound and confidence interval if 500 campers are surveyed? Why?

Exercise 9

In a recent sample of 84 used cars sales costs, the sample mean was $6425 with a standard deviation of $3156. Assume the underlying distribution is approximately normal.

1. Which distribution should you use for this problem? Explain your choice.
2. Define the Random Variable $\overline X$, in words.
3. Construct a 95% confidence interval for the population average cost of a used car.
   1. State the confidence interval.
   2. Sketch the graph.
   3. Calculate the error bound.
4. Explain what a "95% confidence interval" means for this study.

Exercise 10

What is meant by the term "90% confident" when constructing a confidence interval for a mean?

1. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval.
2. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean.
3. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.
4. If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples.

Source: Barbara Illowsky and Susan Dean, https://archive.org/details/CollaborativeStatisticsHomeworkBook/page/n120/mode/1up
This work is licensed under a Creative Commons Attribution 4.0 License.