6.4: Estimating Population Mean
By the end of this chapter, the student should be able to:
- Calculate and interpret confidence intervals for estimating a population mean.
- Calculate the sample size required to estimate a population mean given a desired confidence level and margin of error.
Introduction to Estimating
Suppose you were trying to determine the mean rent of a two-bedroom apartment in your town. You might look in the classified section of the newspaper, write down several rents listed, and average them together. You would have obtained a point estimate of the true mean. If you are trying to determine the percentage of times you make a basket when shooting a basketball, you might count the number of shots you make and divide that by the number of shots you attempted. In this case, you would have obtained a point estimate for the true proportion.
We use sample data to make generalizations about an unknown population. This part of statistics is called inferential statistics . The sample data help us to make an estimate of a population parameter. We realize that the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals.
In this section, you will learn to construct and interpret confidence intervals. Throughout the section, it is important to keep in mind that the confidence interval is a random variable. It is the population parameter that is fixed.
If you worked in the marketing department of an entertainment company, you might be interested in the mean number of songs a consumer downloads a month from iTunes. If so, you could conduct a survey and calculate the sample mean, \(\bar{x}\), and the sample standard deviation, \(s\). You would use \(\bar{x}\) to estimate the population mean and \(s\) to estimate the population standard deviation. The sample mean, \(\bar{x}\), is the point estimate for the population mean, \(\mu\). The sample standard deviation, \(s\), is the point estimate for the population standard deviation, \(\sigma\).
Each of \(\bar{x}\) and \(s\) is called a statistic and each of \(\bar{\mu}\) and \(\sigma\) is called a parameter .
A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers. The interval of numbers is a range of values calculated from a given set of sample data. The confidence interval is likely to include an unknown population parameter.
Suppose for the iTunes example, we do not know the population mean \(\mu\), but we do know that the population standard deviation is \(\sigma = 1\) and our sample size is 100. Then, by the central limit theorem, the standard deviation for the sample mean is
\[\dfrac{\sigma}{\sqrt{n}} = \dfrac{1}{\sqrt{100}} = 0.1.\]
The empirical rule, which applies to bell-shaped distributions, says that in approximately 95% of the samples, the sample mean, \(\bar{x}\), will be within two standard deviations of the population mean \(\mu\). For our iTunes example, two standard deviations is (2)(0.1) = 0.2. The sample mean \(\bar{x}\) is likely to be within 0.2 units of \(\mu\).
Because \(\bar{x}\) is within 0.2 units of \(\mu\), which is unknown, then \(\mu\) is likely to be within 0.2 units of \(\bar{x}\) in 95% of the samples. The population mean \(\mu\) is contained in an interval whose lower number is calculated by taking the sample mean and subtracting two standard deviations (2)(0.1) and whose upper number is calculated by taking the sample mean and adding two standard deviations. In other words, \(\mu\) is between \(\bar{x} - 0.2\) and \(\bar{x} + 0.2\) in 95% of all the samples.
For the iTunes example, suppose that a sample produced a sample mean \(\bar{x} = 2\). Then the unknown population mean \(\mu\) is between
\[\bar{x} - 0.2 = 2 - 0.2 = 1.8\]
and
\[\bar{x} + 0.2 = 2 + 0.2 = 2.2\]
We say that we are 95% confident that the unknown population mean number of songs downloaded from iTunes per month is between 1.8 and 2.2. The 95% confidence interval is (1.8, 2.2). This 95% confidence interval implies two possibilities. Either the interval (1.8, 2.2) contains the true mean \(\mu\) or our sample produced an \(\bar{x}\) that is not within 0.2 units of the true mean \(\mu\). The second possibility happens for only 5% of all the samples (95–100%).
Remember that a confidence interval is created for an unknown population parameter like the population mean, \(\bar{x}\). Confidence intervals for some parameters have the form:
(point estimate – margin of error, point estimate + margin of error)
The margin of error depends on the confidence level or percentage of confidence and the standard error of the mean. When you read newspapers and journals, some reports will use the phrase "margin of error." Other reports will not use that phrase but include a confidence interval as the point estimate plus or minus the margin of error. These are two ways of expressing the same concept.
Although the text only covers symmetrical confidence intervals, there are non-symmetrical confidence intervals (for example, a confidence interval for the standard deviation).
Confidence Intervals for Population Means
To construct a confidence interval for a single unknown population mean \(\mu\), where the population standard deviation is known, we need \(\bar{x}\) as an estimate for \(\mu\) and we need the margin of error. Here, the margin of error (\(EBM\)) is called the error bound for a population mean (abbreviated EBM ). The sample mean \(\bar{x}\) is the point estimate of the unknown population mean \(\mu\).
The confidence interval estimate will have the form:
\[(\text{point estimate} - \text{error bound}, \text{point estimate} + \text{error bound})\nonumber \]
or, in symbols,
\[(\bar{x} - EBM, \bar{x} + EBM)\nonumber \]
The margin of error (\(EBM\)) depends on the confidence level (abbreviated \(CL\)). The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions.
There is another probability called alpha \((\alpha)\). \(\alpha\) is related to the confidence level, \(CL\). \(\alpha\) is the probability that the interval does not contain the unknown population parameter. Mathematically,
\[\alpha + CL = 1.\nonumber \]
Suppose we have collected data from a sample. We know the sample mean but we do not know the mean for the entire population. The sample mean is seven, and the error bound for the mean is 2.5: \(\bar{x} = 7\) and \(EBM = 2.5\)
The confidence interval is (7 – 2.5, 7 + 2.5) and calculating the values gives (4.5, 9.5). If the confidence level (\(CL\)) is 95%, then we say that, "We estimate with 95% confidence that the true value of the population mean is between 4.5 and 9.5."
Suppose we have data from a sample. The sample mean is 15, and the error bound for the mean is 3.2. What is the confidence interval estimate for the population mean?
- Answer
-
(11.8, 18.2)
A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of \(\bar{x} = 10\), and we have constructed the 90% confidence interval (5, 15) where \(EBM = 5\). To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. If we include the central 90%, we leave out a total of \(\alpha = 10%\) in both tails, or 5% in each tail, of the normal distribution.
To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. The value 1.645 is the z -score from a standard normal probability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail.
It is important that the "standard deviation" used must be appropriate for the parameter we are estimating, so in this section we need to use the standard deviation that applies to sample means, which is
\[\dfrac{\sigma}{\sqrt{n}}\nonumber \]
This fraction is commonly called the "standard error of the mean" to distinguish clearly the standard deviation for a mean from the population standard deviation \(\sigma\).
In summary, as a result of the central limit theorem:
- \(\bar{X}\) is normally distributed, that is, \(\bar{X} \sim N(\mu_{X},\dfrac{\sigma}{\sqrt{n}})\).
- When the population standard deviation σ is known, we use a normal distribution to calculate the error bound.
To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are:
- Calculate the sample mean \(\bar{x}\) from the sample data. Remember, in this section we already know the population standard deviation \(\sigma\).
- Find the z -score that corresponds to the confidence level.
- Calculate the error bound \(EBM\).
- Construct the confidence interval.
- Write a sentence that interprets the estimate in the context of the situation in the problem. (Explain what the confidence interval means, in the words of the problem.)
We will first examine each step in more detail, and then illustrate the process with some examples.
Finding the \(z\)-score for the Stated Confidence Level
When we know the population standard deviation \(\sigma\), we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. We need to find the value of \(z\) that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution \(Z \sim N(0, 1)\).
The confidence level, \(CL\), is the area in the middle of the standard normal distribution. \(CL = 1 - \alpha\), so \(\alpha\) is the area that is split equally between the two tails. Each of the tails contains an area equal to \(\dfrac{\alpha}{2}\).
The \(z\)-score that has an area to the right of \(\dfrac{\alpha}{2}\) is denoted by \(z_{\dfrac{\alpha}{2}}\).
For example, when \(CL = 0.95, \alpha = 0.05\) and \(\dfrac{\alpha}{2} = 0.025\); we write \(z_{\dfrac{\alpha}{2}} = z_{0.025}\).
The area to the right of \(z_{0.025}\) is \(0.025\) and the area to the left of \(z_{0.025}\) is \(1 - 0.025 = 0.975\).
\[z_{\dfrac{\alpha}{2}} = z_{0.025} = 1.96\nonumber \]
using a calculator, computer or a standard normal probability table.
There are many online calculators that can be used to compute \(z_\alpha\)'s. For example, you can use this one :
Also, you are encouraged to ask your instructor about which calculator is allowed/recommended for this course.
Use the calculator provided above to verify the following probability statements:
\(z_{0.025}=1.960\)
\(z_{0.05}=1.645\)
\(z_{0.005}=2.576\)
Find \(z_{0.075}\).
Answer
1.44
Calculating the Error Bound
The error bound formula for an unknown population mean \(\mu\) when the population standard deviation \(\sigma\) is known is
\[EBM = z_{\alpha/2} \left(\dfrac{\sigma}{\sqrt{n}}\right)\nonumber \]
There are many online calculators that can be used to compute \(z_\alpha\)'s. For example, you can use this one :
Also, you are encouraged to ask your instructor about which calculator is allowed/recommended for this course.
Use the calculator provided above to verify the following statements:
When \(\alpha=0.1, n=50, \sigma=2.5\) the EBM is \(0.5798\)
When \(\alpha=0.01, n=75, \sigma=13.2\) the EBM is \(3.9324\)
When \(\alpha=0.05, n=10, \sigma=0.7\) the EBM is \(0.4339\)
Find EBM when \(\alpha=0.07, n=32, \sigma=6.7\).
Answer
2.1438
Constructing the Confidence Interval
The confidence interval estimate has the format \((\bar{x} - EBM, \bar{x} + EBM)\).
The graph gives a picture of the entire situation.
\[CL + \dfrac{\alpha}{2} + \dfrac{\alpha}{2} = CL + \alpha = 1.\nonumber \]
There are many online calculators that can be used to compute the confidence intervals. For example, you can use this one :
Also, you are encouraged to ask your instructor about which calculator is allowed/recommended for this course.
Use the calculator provided above to verify the following:
Confidence Level (%): \(90\)
Sample Size: \(47\)
Sample Mean: \(181.7\)
Population Standard Deviation: \(18.9\)
90% Confidence Interval: \((177.1654, 186.2346)\)
Find a 95% confidence interval when the sample size is 30, the sample mean is 60, and the population standard deviation is 5.
Answer
(58.2108, 61.7892)
Writing the Interpretation
The interpretation should clearly state the confidence level (\(CL\)), explain what population parameter is being estimated (here, a population mean ), and state the confidence interval (both endpoints). "We estimate with ___% confidence that the true population mean (include the context of the problem) is between ___ and ___ (include appropriate units)."
Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. A random sample of 36 scores is taken and gives a sample mean (sample mean score) of 68. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).
Find a 90% confidence interval for the true (population) mean of statistics exam scores.
Answer
To find the confidence interval, you need the sample mean, \(\bar{x}\), and the \(EBM\).
\[\bar{x} = 68\nonumber \]
\[EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right)\nonumber \]
\[\sigma = 3; n = 36\nonumber \]
The confidence level is 90% ( CL = 0.90)
\[CL = 0.90\nonumber \]
so
\[\alpha = 1 – CL = 1 – 0.90 = 0.10\nonumber \]
\[\dfrac{\alpha}{2} = 0.05 z_{\dfrac{\alpha}{2}} = z_{0.05}\nonumber \]
The area to the right of \(z_{0.05}\) is \(0.05\) and the area to the left of \(z_{0.05}\) is \(1 - 0.05 = 0.95\).
\[z_{\dfrac{\alpha}{2}} = z_{0.05} = 1.645\nonumber \]
using \(\text{invNorm}(0.95, 0, 1)\) on the TI-83,83+, and 84+ calculators. This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution.
\[EBM = (1.645)\left(\dfrac{3}{\sqrt{36}}\right) = 0.8225\nonumber \]
\[\bar{x} - EBM = 68 - 0.8225 = 67.1775\nonumber \]
\[\bar{x} + EBM = 68 + 0.8225 = 68.8225\nonumber \]
The 90% confidence interval is (67.1775, 68.8225).
Interpretation: We estimate with 90% confidence that the true population mean exam score for all statistics students is between 67.18 and 68.82.
Explanation of 90% Confidence Level: Ninety percent of all confidence intervals constructed in this way contain the true mean statistics exam score. For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score.
Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes. Find a 90% confidence interval estimate for the population mean delivery time.
- Answer
-
(34.1347, 37.8653)
The Specific Absorption Rate (SAR) for a cell phone measures the amount of radio frequency (RF) energy absorbed by the user’s body when using the handset. Every cell phone emits RF energy. Different phone models have different SAR measures. To receive certification from the Federal Communications Commission (FCC) for sale in the United States, the SAR level for a cell phone must be no more than 1.6 watts per kilogram. Table shows the highest SAR level for a random selection of cell phone models as measured by the FCC.
| Phone Model | SAR | Phone Model | SAR | Phone Model | SAR |
|---|---|---|---|---|---|
| Apple iPhone 4S | 1.11 | LG Ally | 1.36 | Pantech Laser | 0.74 |
| BlackBerry Pearl 8120 | 1.48 | LG AX275 | 1.34 | Samsung Character | 0.5 |
| BlackBerry Tour 9630 | 1.43 | LG Cosmos | 1.18 | Samsung Epic 4G Touch | 0.4 |
| Cricket TXTM8 | 1.3 | LG CU515 | 1.3 | Samsung M240 | 0.867 |
| HP/Palm Centro | 1.09 | LG Trax CU575 | 1.26 | Samsung Messager III SCH-R750 | 0.68 |
| HTC One V | 0.455 | Motorola Q9h | 1.29 | Samsung Nexus S | 0.51 |
| HTC Touch Pro 2 | 1.41 | Motorola Razr2 V8 | 0.36 | Samsung SGH-A227 | 1.13 |
| Huawei M835 Ideos | 0.82 | Motorola Razr2 V9 | 0.52 | SGH-a107 GoPhone | 0.3 |
| Kyocera DuraPlus | 0.78 | Motorola V195s | 1.6 | Sony W350a | 1.48 |
| Kyocera K127 Marbl | 1.25 | Nokia 1680 | 1.39 | T-Mobile Concord | 1.38 |
Find a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. Assume that the population standard deviation is \(\sigma = 0.337\).
Solution
To find the confidence interval, start by finding the point estimate: the sample mean.
\[\bar{x} = 1.024\nonumber \]
Next, find the \(EBM\). Because you are creating a 98% confidence interval, \(CL = 0.98\).
You need to find \(z_{0.01}\) having the property that the area under the normal density curve to the right of \(z_{0.01}\) is \(0.01\) and the area to the left is 0.99. Use your calculator, a computer, or a probability table for the standard normal distribution to find \(z_{0.01} = 2.326\).
\(EBM = (z_{0.01})\dfrac{\sigma}{\sqrt{n}} = (2.326)\dfrac{0.337}{\sqrt{30}} =0.1431\)
To find the 98% confidence interval, find \(\bar{x} \pm EBM\).
\(\bar{x} - EBM = 1.024 – 0.1431 = 0.8809\)
\(\bar{x} - EBM = 1.024 – 0.1431 = 1.1671\)
We estimate with 98% confidence that the true SAR mean for the population of cell phones in the United States is between 0.8809 and 1.1671 watts per kilogram.
Table shows a different random sampling of 20 cell phone models. Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. As previously, assume that the population standard deviation is \(\sigma = 0.337\).
| Phone Model | SAR | Phone Model | SAR |
|---|---|---|---|
| Blackberry Pearl 8120 | 1.48 | Nokia E71x | 1.53 |
| HTC Evo Design 4G | 0.8 | Nokia N75 | 0.68 |
| HTC Freestyle | 1.15 | Nokia N79 | 1.4 |
| LG Ally | 1.36 | Sagem Puma | 1.24 |
| LG Fathom | 0.77 | Samsung Fascinate | 0.57 |
| LG Optimus Vu | 0.462 | Samsung Infuse 4G | 0.2 |
| Motorola Cliq XT | 1.36 | Samsung Nexus S | 0.51 |
| Motorola Droid Pro | 1.39 | Samsung Replenish | 0.3 |
| Motorola Droid Razr M | 1.3 | Sony W518a Walkman | 0.73 |
| Nokia 7705 Twist | 0.7 | ZTE C79 | 0.869 |
Answer
\[\bar{x} = 0.940\nonumber \]
\[\dfrac{\alpha}{2} = \dfrac{1 - CL}{2} = \dfrac{1 - 0.93}{2} = 0.035\nonumber \]
\[z_{0.035} = 1.812\nonumber \]
\[EBM = (z_{0.035})\left(\dfrac{\sigma}{\sqrt{n}}\right) = (1.812)\left(\dfrac{0.337}{\sqrt{20}}\right) = 0.1365\nonumber \]
\[\bar{x} - EBM = 0.940 - 0.1365 = 0.8035\nonumber \]
\[\bar{x} + EBM = 0.940 + 0.1365 = 1.0765\nonumber \]
We estimate with 93% confidence that the true SAR mean for the population of cell phones in the United States is between 0.8035 and 1.0765 watts per kilogram.
Notice the difference in the confidence intervals calculated in Example and the following Exercise. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. Even though the intervals are different, they do not yield conflicting information. The effects of these kinds of changes are the subject of the next section in this chapter.
Changing the Confidence Level or Sample Size
Suppose we change the original problem in Example by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score.
Answer
To find the confidence interval, you need the sample mean, \(\bar{x}\), and the \(EBM\).
\(\bar{x} = 68\)
\(EBM = \left(z_{\dfrac{\alpha}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right)\)
\(\sigma = 3; n = 36\); The confidence level is 95% ( CL = 0.95).
\(CL = 0.95\) so \(\alpha = 1 – CL = 1 – 0.95 = 0.05\)
\(\dfrac{\alpha}{2} = 0.025 z_{\dfrac{\alpha}{2}} = z_{0.025}\)
The area to the right of \(z_{0.025}\) is 0.025 and the area to the left of \(z_{0.025}\) is \(1 – 0.025 = 0.975\).
\(z_{\dfrac{\alpha}{2}} = z_{0.025} = 1.96\)
when using invnorm(0.975,0,1) on the TI-83, 83+, or 84+ calculators. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution.)
\(EBM = (1.96)\left(\dfrac{3}{\sqrt{36}}\right) = 0.98\)
\(\bar{x} – EBM = 68 – 0.98 = 67.02\)
\(\bar{x} + EBM = 68 + 0.98 = 68.98\)
Notice that the \(EBM\) is larger for a 95% confidence level in the original problem.
Interpretation
We estimate with 95% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98.
Explanation of 95% Confidence Level
Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score.
Comparing the results
The 90% confidence interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider. To be more confident that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, the confidence interval necessarily needs to be wider.
Summary: Effect of Changing the Confidence Level
- Increasing the confidence level increases the error bound, making the confidence interval wider.
- Decreasing the confidence level decreases the error bound, making the confidence interval narrower.
Calculating the Sample Size \(n\)
If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. The error bound formula for a population mean when the population standard deviation is known is
\[EBM = \left(z_{\dfrac{a}{2}}\right)\left(\dfrac{\sigma}{\sqrt{n}}\right) \label{samplesize}\nonumber \]
The formula for sample size is \(n = \dfrac{z^{2}\sigma^{2}}{EBM^{2}}\), found by solving the error bound formula for \(n\). In the formulas, \(z\) is \(z_{\dfrac{a}{2}}\), corresponding to the desired confidence level. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.
The population standard deviation for the age of Foothill College students is 15 years. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?
Solution
- From the problem, we know that \(\sigma = 15\) and \(EBM = 2\).
- \(z = z_{0.025} = 1.96\), because the confidence level is 95%.
- \(n = \dfrac{z^{2}\sigma^{2}}{EBM^{2}} = \dfrac{(1.96)^{2}(15)^{2}}{2^{2}}\) using the sample size equation.
- Use \(n = 217\): Always round the answer UP to the next higher integer to ensure that the sample size is large enough.
Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students.
The population standard deviation for the height of high school basketball players is three inches. If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed?
- Answer
-
35 students