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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. To construct a confidence interval for a single unknown population mean [latex]\mu[/latex], where the population standard deviation is known, we need [latex]\overline{x}[/latex], which is the point estimate of the unknown population mean [latex]\mu[/latex]. The confidence interval estimate will have the form: [latex]\begin{eqnarray*} \mbox{Lower Limit} & = & \overline{x}-\mbox{margin of error} \\ \\ \mbox{Upper Limit} & = & \overline{x}-\mbox{margin of error} \end{eqnarray*}[/latex] The margin of error depends on the confidence level. 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 their conclusions. Watch this video: Confidence Intervals – Introduction by Joshua Emmanual [3:34] Suppose we have collected data from a sample. The sample mean is 7 and the margin of error is 2.5. The confidence interval is: [latex]\begin{eqnarray*} \mbox{Lower Limit} & = & 7 -2.5=4.5 \\ \\ \mbox{Upper Limit} & = & 7+2.5 =9.5 \end{eqnarray*}[/latex] If the confidence level 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 margin of error is 3.2. What is the confidence interval estimate for the population mean? Click to see Solution[latex]\begin{eqnarray*} \mbox{Lower Limit} & = & 15 -3.2=11.8 \\ \\ \mbox{Upper LImit} & = & 15+3.2 =18.2 \end{eqnarray*}[/latex] 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 [latex]\overline{x}=10[/latex] and we have constructed the 90% confidence interval with a lower limit of 5 and an upper limit of 15. 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 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 [latex]z[/latex]-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 [latex]\displaystyle{\frac{\sigma}{\sqrt{n}}}[/latex] (the standard deviation of the sample means). The fraction [latex]\displaystyle{\frac{\sigma}{\sqrt{n}}}[/latex] is commonly called the standard error of the mean in order to clearly distinguish the standard deviation for a sample mean from the population standard deviation [latex]\sigma[/latex]. Calculating the Confidence IntervalTo 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:
We will first examine each step in more detail, and then illustrate the process with some examples. Finding the [latex]z[/latex]-score for the Confidence LevelWhen we know the population standard deviation [latex]\sigma[/latex], we use a standard normal distribution to calculate the margin of error and construct the confidence interval. We need to find the value of [latex]z[/latex] that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution. The confidence level [latex]C[/latex] is the area in the middle of the standard normal distribution. The remaining area, [latex]1-C[/latex], is split equally between the two tails, so each of the tails contains an area equal to [latex]\displaystyle{\frac{1-C}{2}}[/latex]. The [latex]z[/latex]-score needed to construct the confidence interval is the [latex]z[/latex]-score so that the entire area to the left of [latex]z[/latex]-score equals the area in the middle (the confidence level) plus the area in the left tail [latex]\displaystyle{\left(\frac{1-C}{2}\right)}[/latex]. That is, the required [latex]z[/latex]-score for the confidence interval is the [latex]z[/latex]-score so that the entire area to the left of the [latex]z[/latex]-score is [latex]\displaystyle{C+\frac{1-C}{2}}[/latex] For example, if the confidence level is 95%, then the area in the center of the standard normal distribution is 0.95 and the area in the left tail is [latex]\displaystyle{\frac{1-0.95}{2}=0.025}[/latex]. We would need to find the [latex]z[/latex]-score so that the entire area to the left of the [latex]z[/latex]-score equals [latex]0.95+0.025=0.975[/latex]. To find the [latex]z[/latex]-score to construct a confidence interval with confidence level [latex]C[/latex], use the norm.s.inv(area to the left of z) function.
The output from the norm.s.inv function is the value of [latex]z[/latex]-score needed to construct the confidence interval. NOTEThe norm.s.inv function requires that we enter the entire area to the left of the unknown [latex]z[/latex]-score. This area includes the confidence level (the area in the middle of the distribution) plus the remaining area in the left tail. Calculating the Margin of ErrorThe margin of error for a confidence interval with confidence level [latex]C[/latex] for an unknown population mean [latex]\mu[/latex] when the population standard deviation [latex]\sigma[/latex] is known is [latex]\displaystyle{\mbox{Margin of Error}=z \times \frac{\sigma}{\sqrt{n}}}[/latex] where [latex]z[/latex] is the the [latex]z[/latex]-score so the area the left of [latex]z[/latex] is [latex]\displaystyle{C+\frac{1-C}{2}}[/latex]. Constructing the Confidence IntervalThe limits for the confidence interval with confidence level [latex]C[/latex] for an unknown population mean [latex]\mu[/latex] when the population standard deviation [latex]\sigma[/latex] is known are [latex]\begin{eqnarray*} \mbox{Lower Limit} & = & \overline{x}-z \times \frac{\sigma}{\sqrt{n}} \\ \\ \mbox{Upper Limit} & = & \overline{x}+z \times \frac{\sigma}{\sqrt{n}} \end{eqnarray*}[/latex] where [latex]z[/latex] is the [latex]z[/latex]-score so the area the left of [latex]z[/latex] is [latex]\displaystyle{C+\frac{1-C}{2}}[/latex]. Interpreting a Confidence IntervalThe interpretation should clearly state the confidence level [latex]C[/latex], explain what population parameter is being estimated (in this case 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).” Watch
this video: Confidence Interval for a population mean – [latex]\sigma[/latex] known by Joshua Emmanuel [4:30] Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of 3 points. A random sample of 36 scores is taken and has a sample mean of 68 points.
Solution:
NOTES
Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of 6 minutes. A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes.
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. This table shows the highest SAR level for a random selection of cell phone models as measured by the FCC.
Solution:
This table shows a different random sampling of 20 cell phone models. As previously, assume that the population standard deviation is [latex]\sigma = 0.337[/latex].
Notice the difference in the confidence intervals calculated in the Example and Try It just completed. 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. Changing the Confidence LevelSuppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of 3 points. A random sample of 36 scores is taken and gives a sample mean of 68 points. Previously we found a 90% confidence interval for the mean exam score. Now, find a 95% confidence interval for the mean exam score. Interpret the 95% confidence interval. Solution: To find the confidence interval, we need to find the [latex]z[/latex]-score for the 95% confidence interval. This means that we need to find the [latex]z[/latex]-score so that the entire area to the left of [latex]z[/latex] is [latex]\displaystyle{0.95+\frac{1-0.95}{2}=0.975}[/latex].
So [latex]z=1.9599....[/latex]. From the question [latex]\overline{x}=68[/latex], [latex]\sigma=3[/latex] and [latex]n=36[/latex]. The 95% confidence interval is [latex]\begin{eqnarray*}\\ \mbox{Lower Limit} & = & \overline{x}-z \times \frac{\sigma}{\sqrt{n}} \\ & = & 68-1.9599... \times \frac{3}{\sqrt{36}} \\ & = & 67.02 \\ \\ \mbox{Upper Limit} & = & \overline{x}+z \times \frac{\sigma}{\sqrt{n}} \\ & = & 68+1.9599... \times \frac{3}{\sqrt{36}} \\ & = & 68.98 \\ \\ \end{eqnarray*}[/latex] We are 95% confident that the mean exam score is between 67.02 points and 68.98 points. Comparing the ResultsFor the exam scores examples, the 90% confidence interval has a lower limit of 67.18 and an upper limit of 68.82, and the 95% confidence interval has a lower limit of 67.02 and an upper limit of 68.98. Notice that the 95% confidence interval is wider (the distance between the limits is larger in the 95% confidence interval). If we 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. Effect of Changing the Confidence Level
Changing the Sample SizeSuppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of 3 points. Previously, we found a 90% confidence interval for the mean exam score using a sample of size 36 with a sample mean of 68.
Solution:
Comparing the ResultsFor the exam scores examples, the 90% confidence interval with a sample size of 36 has a lower limit of 67.18 and an upper limit of 68.82, with a sample size of 100 has a lower limit is 67.51 and an upper limit is 68.49, and with a sample size of 25 has a lower limit is 67.01 and an upper limit is 69.27. When the sample size increased, the confidence interval is narrower. When the sample size decreased, the confidence interval is wider. Generally, the smaller the sample size, the wider the confidence interval needs to be in order to achieve the same level of confidence. Effect of Changing the Sample Size
Concept ReviewIn this section, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known. A confidence interval has the general form: [latex]\begin{eqnarray*}\\ \mbox{Lower Limit} & = & \overline{x}-\mbox{margin of error} \\ \\ \mbox{Upper Limit} & = & \overline{x}-\mbox{margin of error}\\ \\ \end{eqnarray*}[/latex] The general form for a confidence interval for a single population mean, known standard deviation is given by [latex]\begin{eqnarray*}\\ \mbox{Lower Limit} & = & \overline{x}-z \times \frac{\sigma}{\sqrt{n}} \\ \\ \mbox{Upper Limit} & = & \overline{x}+z \times \frac{\sigma}{\sqrt{n}}\\ \\ \end{eqnarray*}[/latex] where [latex]z[/latex] is the the [latex]z[/latex]-score so the area the left of [latex]z[/latex] is [latex]\displaystyle{C+\frac{1-C}{2}}[/latex]. The calculation of the margin of error depends on the size of the sample and the level of confidence required. 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 margin of error increases as well. As the sample size increases, the margin of error decreases. Attribution“8.1 A Single Population Mean using the Normal Distribution“ in Introductory Statistics by OpenStax is licensed under a Creative Commons Attribution 4.0 International License. What is a 95% CI of population mean?A 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. This is not the same as a range that contains 95% of the values. The graph below emphasizes this distinction. The graph shows three samples (of different size) all sampled from the same population.
What is the confidence interval for 95% confidence level?88 – (1.96 x 0.53) = 86.96 mmHg. This is called the 95% confidence interval , and we can say that there is only a 5% chance that the range 86.96 to 89.04 mmHg excludes the mean of the population. If we take the mean plus or minus three times its standard error, the range would be 86.41 to 89.59.
What is the 99% confidence interval for the population mean?zc. Common Levels of Confidence. If the level of confidence is 99%, this means that we are 99% confident that the interval contains the population mean, µ. The corresponding z-scores are ± 2.575.
How would you interpret a 95% confidence interval for the mean?A 95% confidence interval (CI) of the mean is a range with an upper and lower number calculated from a sample. Because the true population mean is unknown, this range describes possible values that the mean could be.
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