8.S: Confidence Intervals (Summary)

Chapter Review

In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. When estimating a population mean, the margin of error is called the error bound for a population mean (EBM). A confidence interval has the general form:

$$(\text{lower bound, upper bound}) = (\text{point estimate} – EBM, \text{point estimate} + EBM)$$

The calculation of $$EBM$$ depends on the size of the sample and the level of confidence desired. The confidence level is the percent of all possible samples that can be expected to include the true population parameter. As the confidence level increases, the corresponding $$EBM$$ increases as well. As the sample size increases, the $$EBM$$ decreases. By the central limit theorem,

$$EBM = z\frac{\sigma}{\sqrt{n}}$$

Given a confidence interval, you can work backwards to find the error bound ($$EBM$$) or the sample mean. To find the error bound, find the difference of the upper bound of the interval and the mean. If you do not know the sample mean, you can find the error bound by calculating half the difference of the upper and lower bounds. To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean.

Sometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. In that case, solve the $$EBM$$ formula for $$n$$ to discover the size of the sample that is needed to achieve this goal:

$$n = \frac{z^{2}\sigma^{2}}{EBM^{2}}$$

Formula Review

$$\bar{X} - N \left(\mu_{x}, \frac{\sigma}{\sqrt{n}}\right)$$ The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size.

The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by

$(\text{lower bound, upper bound}) = (\text{point estimate} – EBM, \text{point estimate} + EBM)$

$= \bar{x} - EBM, \bar{x} + EBM$

$= \left(\bar{x} - z\frac{\sigma}{\sqrt{n}}, \bar{x} + z\frac{\sigma}{\sqrt{n}}\right)$

$$EBM = z\frac{\sigma}{\sqrt{n}} =$$ the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known.

$$CL =$$ confidence level, or the proportion of confidence intervals created that are expected to contain the true population parameter

$$\alpha = 1 – CL =$$ the proportion of confidence intervals that will not contain the population parameter

$$z_{\frac{\alpha}{2}}$$ = the $$z$$-score with the property that the area to the right of the z-score is $$\frac{\propto}{2}$$ this is the $$z$$-score used in the calculation of "$$EBM$$ where $$\alpha = 1 – CL$$".

$$n = \frac{z^{2}\sigma^{2}}{EBM^{2}}$$ the formula used to determine the sample size ($$n$$) needed to achieve a desired margin of error at a given level of confidence

General form of a confidence interval

$(\text{lower value, upper value}) = (\text{point estimate} - \text{error bound, point estimate} + \text{error bound})$

To find the error bound when you know the confidence interval

$\text{error bound} = \text{upper value} - \text{point estimate}$ OR $\text{error bound} = \frac{\text{upper value - lower value}}{2}$

Single Population Mean, Known Standard Deviation, Normal Distribution

Use the Normal Distribution for Means, Population Standard Deviation is Known $$EBM = z\frac{\alpha}{2} \cdot \frac{\sigma}{\sqrt{n}}$$

The confidence interval has the format $$(\bar{x} - EBM, \bar{x} + EBM)$$.

Use the following information to answer the next five exercises: The standard deviation of the weights of elephants is known to be approximately 15 pounds. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 pounds. The sample standard deviation is 11 pounds.

Exercise 8.2.8

Identify the following:

1. $$\bar{x} =$$ _____
2. $$\sigma =$$ _____
3. $$n =$$ _____

1. 244
2. 15
3. 50

Exercise 8.2.9

In words, define the random variables $$X$$ and $$\bar{X}$$.

Exercise 8.2.10

Which distribution should you use for this problem?

$$N\left(244, \frac{15}{\sqrt{50}}\right)$$

Exercise 8.2.11

Construct a 95% confidence interval for the population mean weight of newborn elephants. State the confidence interval, sketch the graph, and calculate the error bound.

Exercise 8.2.12

What will happen to the confidence interval obtained, if 500 newborn elephants are weighed instead of 50? Why?

As the sample size increases, there will be less variability in the mean, so the interval size decreases.

Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal.

Exercise 8.2.13

Identify the following:

1. $$\bar{x} =$$ _____
2. $$\sigma =$$ _____
3. $$n =$$ _____

Exercise 8.2.14

In words, define the random variables $$X$$ and $$\bar{X}$$.

$$X$$ is the time in minutes it takes to complete the U.S. Census short form. $$\bar{X}$$ is the mean time it took a sample of 200 people to complete the U.S. Census short form.

Exercise 8.2.15

Which distribution should you use for this problem?

Exercise 8.2.16

Construct a 90% confidence interval for the population mean time to complete the forms. State the confidence interval, sketch the graph, and calculate the error bound.

$$CI: (7.9441, 8.4559)$$ Figure 8.2.3.

$$EBM = 0.26$$

Exercise 8.2.17

If the Census wants to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?

Exercise 8.2.18

If the Census did another survey, kept the error bound the same, and surveyed only 50 people instead of 200, what would happen to the level of confidence? Why?

The level of confidence would decrease because decreasing $$n$$ makes the confidence interval wider, so at the same error bound, the confidence level decreases.

Exercise 8.2.19

Suppose the Census needed to be 98% confident of the population mean length of time. Would the Census have to survey more people? Why or why not?

Use the following information to answer the next ten exercises: A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 pounds with a standard deviation of 0.1 pounds. The population standard deviation is known to be 0.2 pounds.

Exercise 8.2.20

Identify the following:

1. $$\bar{x} =$$ _____
2. $$\sigma =$$ _____
3. $$n =$$ _____

1. $$\bar{x} = 2.2$$
2. $$\sigma = 0.2$$
3. $$n = 20$$

Exercise 8.2.21

In words, define the random variable $$X$$.

Exercise 8.2.22

In words, define the random variable $$\bar{X}$$.

$$\bar{X}$$ is the mean weight of a sample of 20 heads of lettuce.

Exercise 8.2.23

Which distribution should you use for this problem?

Exercise 8.2.24

Construct a 90% confidence interval for the population mean weight of the heads of lettuce. State the confidence interval, sketch the graph, and calculate the error bound.

$$EBM = 0.07$$

$$CI: (2.1264, 2.2736)$$ Figure 8.2.4.

Exercise 8.2.25

Construct a 95% confidence interval for the population mean weight of the heads of lettuce. State the confidence interval, sketch the graph, and calculate the error bound.

Exercise 8.2.26

In complete sentences, explain why the confidence interval in Exercise is larger than in Exercise.

The interval is greater because the level of confidence increased. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution. Therefore, a larger confidence level results in larger areas and larger intervals.

Exercise 8.2.27

In complete sentences, give an interpretation of what the interval in Exercise means.

Exercise 8.2.28

What would happen if 40 heads of lettuce were sampled instead of 20, and the error bound remained the same?

The confidence level would increase.

Exercise 8.2.29

What would happen if 40 heads of lettuce were sampled instead of 20, and the confidence level remained the same?

Use the following information to answer the next 14 exercises: The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. Let $$X$$ = the age of a Winter Foothill College student.

Exercise 8.2.30

$$\bar{x} =$$ _____

30.4

Exercise 8.2.31

$$n =$$ _____

Exercise 8.2.32

________ $$= 15$$

$$\sigma$$

Exercise 8.2.33

In words, define the random variable $$\bar{X}$$.

Exercise 8.2.34

What is $$\bar{x}$$ estimating?

$$\mu$$

Exercise 8.2.35

Is $$\sigma_{x}$$ known?

Exercise 8.2.36

As a result of your answer to Exercise, state the exact distribution to use when calculating the confidence interval.

normal

Construct a 95% Confidence Interval for the true mean age of Winter Foothill College students by working out then answering the next seven exercises.

Exercise 8.2.37

How much area is in both tails (combined)? $$\alpha =$$ ________

Exercise 8.2.38

How much area is in each tail? $$\frac{\alpha}{2} =$$ ________

0.025

Exercise 8.2.39

Identify the following specifications:

1. lower limit
2. upper limit
3. error bound

Exercise 8.2.40

The 95% confidence interval is:__________________.

(24.52,36.28)

Exercise 8.2.41

Fill in the blanks on the graph with the areas, upper and lower limits of the confidence interval, and the sample mean. Figure 8.2.5.

Exercise 8.2.42

In one complete sentence, explain what the interval means.

We are 95% confident that the true mean age for Winger Foothill College students is between 24.52 and 36.28.

Exercise 8.2.43

Using the same mean, standard deviation, and level of confidence, suppose that $$n$$ were 69 instead of 25. Would the error bound become larger or smaller? How do you know?

Exercise 8.2.44

Using the same mean, standard deviation, and sample size, how would the error bound change if the confidence level were reduced to 90%? Why?

The error bound for the mean would decrease because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.

Chapter Review

In many cases, the researcher does not know the population standard deviation, $$\sigma$$, of the measure being studied. In these cases, it is common to use the sample standard deviation, $$s$$, as an estimate of $$\sigma$$. The normal distribution creates accurate confidence intervals when $$\sigma$$ is known, but it is not as accurate when $$s$$ is used as an estimate. In this case, the Student’s t-distribution is much better. Define a t-score using the following formula:

$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$

The $$t$$-score follows the Student’s $$t$$-distribution with $$n – 1$$ degrees of freedom. The confidence interval under this distribution is calculated with $$EBM = \left(t_{\frac{\alpha}{2}}\right)\frac{s}{\sqrt{n}}$$ where $$t_{\frac{\alpha}{2}}$$ is the $$t$$-score with area to the right equal to $$\frac{\alpha}{2}$$, $$s$$ is the sample standard deviation, and $$n$$ is the sample size. Use a table, calculator, or computer to find $$t_{\frac{\alpha}{2}}$$ for a given $$\alpha$$.

Formula Review

$$s =$$ the standard deviation of sample values.

$$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$$ is the formula for the $$t$$-score which measures how far away a measure is from the population mean in the Student’s $$t$$-distribution

$$df = n - 1$$; the degrees of freedom for a Student’s $$t$$-distribution where n represents the size of the sample

$$T \sim t_{df}$$ the random variable, $$T$$, has a Student’s $$t$$-distribution with $$df$$ degrees of freedom

$$EBM = t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}} =$$ the error bound for the population mean when the population standard deviation is unknown

$$t_{\frac{\alpha}{2}}$$ is the $$t$$-score in the Student’s $$t$$-distribution with area to the right equal to $$\frac{\alpha}{2}$$

The general form for a confidence interval for a single mean, population standard deviation unknown, Student's t is given by (lower bound, upper bound)

$= (\text{point estimate} – EBM, \text{point estimate} + EBM)$

$= \left(\bar{x} - \frac{ts}{\sqrt{n}}, \bar{x} + \frac{ts}{\sqrt{n}}\right)$

Use the following information to answer the next five exercises. A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed 70 patients. The sample mean was 1.5 hours with a sample standard deviation of 0.5 hours.

Exercise 8.3.3

Identify the following:

1. $$\bar{x} =$$_______
2. $$s_{x} =$$_______
3. $$n =$$_______
4. $$n - 1=$$_______

Exercise 8.3.4

Define the random variables $$X$$ and $$\bar{X}$$ in words.

$$X$$ is the number of hours a patient waits in the emergency room before being called back to be examined. $$\bar{X}$$ is the mean wait time of 70 patients in the emergency room.

Exercise 8.3.5

Which distribution should you use for this problem?

Exercise 8.3.6

Construct a 95% confidence interval for the population mean time spent waiting. State the confidence interval, sketch the graph, and calculate the error bound.

$$CI: (1.3808, 1.6192)$$ Figure 8.3.1.

$$EBM = 0.12$$

Exercise 8.3.7

Explain in complete sentences what the confidence interval means.

Use the following information to answer the next six exercises: One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watched an average of 151 hours each month with a standard deviation of 32 hours. Assume that the underlying population distribution is normal.

Exercise 8.3.8

1. $$\bar{x} =$$_______
2. $$s_{x} =$$_______
3. $$n =$$_______
4. $$n - 1=$$_______

1. $$\bar{x} = 151$$
2. $$s_{x} = 32$$
3. $$n = 108$$
4. $$n - 1= 107$$

Exercise 8.3.9

Define the random variable $$X$$ in words.

Exercise 8.3.10

Define the random variable $$\bar{X}$$ in words.

$$\bar{X}$$ is the mean number of hours spent watching television per month from a sample of 108 Americans.

Exercise 8.3.11

Which distribution should you use for this problem?

Exercise 8.3.12

Construct a 99% confidence interval for the population mean hours spent watching television per month. (a) State the confidence interval, (b) sketch the graph, and (c) calculate the error bound.

$$CI: (142.92, 159.08)$$ Figure 8.3.2.

$$EBM = 8.08$$

Exercise 8.3.13

Why would the error bound change if the confidence level were lowered to 95%?

Use the following information to answer the next 13 exercises: The data in Table are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let $$X =$$ the number of colors on a national flag.

$$X$$ Freq.
1 1
2 7
3 18
4 7
5 6

Exercise 8.3.14

1. $$\bar{x} =$$_______
2. $$s_{x} =$$_______
3. $$n =$$_______

1. 3.26
2. 1.02
3. 39

Exercise 8.3.15

Define the random variable $$\bar{X}$$ in words.

Exercise 8.3.16

What is $$\bar{x}$$ estimating?

$$\mu$$

Exercise 8.3.17

Is $$\sigma_{x}$$ known?

Exercise 8.3.18

As a result of your answer to Exercise, state the exact distribution to use when calculating the confidence interval.

$$t_{38}$$

Construct a 95% confidence interval for the true mean number of colors on national flags.

Exercise 8.3.19

How much area is in both tails (combined)?

Exercise 8.3.20

How much area is in each tail?

0.025

Exercise 8.3.21

Calculate the following:

1. lower limit
2. upper limit
3. error bound

Exercise 8.3.22

The 95% confidence interval is_____.

(2.93, 3.59)

Exercise 8.3.23

Fill in the blanks on the graph with the areas, the upper and lower limits of the Confidence Interval and the sample mean. Figure 8.3.3.

Exercise 8.3.24

In one complete sentence, explain what the interval means.

We are 95% confident that the true mean number of colors for national flags is between 2.93 colors and 3.59 colors.

Exercise 8.3.25

Using the same $$\bar{x}$$, $$s_{x}$$, and level of confidence, suppose that $$n$$ were 69 instead of 39. Would the error bound become larger or smaller? How do you know?

The error bound would become $$EBM = 0.245$$. This error bound decreases because as sample sizes increase, variability decreases and we need less interval length to capture the true mean.

Exercise 8.3.26

Using the same $$\bar{x}$$, $$s_{x}$$, and $$n = 39$$, how would the error bound change if the confidence level were reduced to 90%? Why?

References

1. Jensen, Tom. “Democrats, Republicans Divided on Opinion of Music Icons.” Public Policy Polling. Available online at http://www.publicpolicypolling.com/Day2MusicPoll.pdf (accessed July 2, 2013).
2. Madden, Mary, Amanda Lenhart, Sandra Coresi, Urs Gasser, Maeve Duggan, Aaron Smith, and Meredith Beaton. “Teens, Social Media, and Privacy.” PewInternet, 2013. Available online at http://www.pewinternet.org/Reports/2...d-Privacy.aspx (accessed July 2, 2013).
3. Prince Survey Research Associates International. “2013 Teen and Privacy Management Survey.” Pew Research Center: Internet and American Life Project. Available online at http://www.pewinternet.org/~/media//...al%20Media.pdf (accessed July 2, 2013).
4. Saad, Lydia. “Three in Four U.S. Workers Plan to Work Pas Retirement Age: Slightly more say they will do this by choice rather than necessity.” Gallup® Economy, 2013. Available online at http://www.gallup.com/poll/162758/th...ement-age.aspx (accessed July 2, 2013).
5. The Field Poll. Available online at http://field.com/fieldpollonline/subscribers/ (accessed July 2, 2013).
6. Zogby. “New SUNYIT/Zogby Analytics Poll: Few Americans Worry about Emergency Situations Occurring in Their Community; Only one in three have an Emergency Plan; 70% Support Infrastructure ‘Investment’ for National Security.” Zogby Analytics, 2013. Available online at http://www.zogbyanalytics.com/news/2...analytics-poll (accessed July 2, 2013).
7. “52% Say Big-Time College Athletics Corrupt Education Process.” Rasmussen Reports, 2013. Available online at http://www.rasmussenreports.com/publ...cation_process (accessed July 2, 2013).