Module 28 - Estimating a Population Mean (1 of 3) In "Estimating a Population Mean," we focus on how to use a sample mean to estimate a population mean. This is the type of thinking we did in Modules 7 and 8 when we used a sample proportion to estimate a population proportion. Let s take a moment to review what we learned in the modules Linking Probability to Statistical Inference and Inference for One Proportion, and then we ll see how it relates to the current module. In Linking Probability to Statistical Inference, we noted that random samples vary, so we expect to see variability in sample proportions. In the section "Distribution of Sample Means" in that module, we made the same observations about sample means. In both cases, a normal model is a good fit for the sampling distribution when appropriate conditions are met. We also noted in that module that a sample proportion is an estimate for the population proportion. We do not expect the sample proportion to equal the population proportion, so there is some error. The error is due to random chance. Likewise, a sample mean is an estimate for the population mean, but there will be some error due to random chance.
Comment The following material comes from Concepts in Statistics written by the Open Learning Initiative (OLI) Recall that, in Inference for One Proportion, we adjusted the standard error by replacing p with the sample proportion. Doing so made sense because the goal of the confidence interval is to estimate p. So the margin of error in the confidence interval formula changed. Here is the adjusted formula. We will eventually have to adjust the standard error for the sampling distribution of sample means, too. It makes sense because in many situations we will not know the population standard deviation, σ. This adjustment is more complicated than the adjustment to standard error for sample proportions, so before we do it, let s practice finding the confidence interval for µ assuming we know σ.this adjustment changed the normality conditions. We use this adjusted confidence interval to estimate pwhen the successes and failures in the actual sample are at least 10. Assuming we know σ is realistic when a lot of previous research has been done. For example, when we are estimating height, weight, or scores on a standardized test, previous research gives us reliable values for σ. EXAMPLE Estimating Mean SAT Math Score The SAT is the most widely used college admission exam. (Most community colleges do not require students to take this exam.) The mean SAT math score varies by state and by year, so the value of µ depends on the state and the year. But let s assume that the shape and spread of the distribution of individual SAT math scores in each state is the same each year. More specifically, assume that individual SAT math scores consistently have a normal distribution with a standard deviation of 100. An educational researcher wants to estimate the mean SAT math score (μ) for his state this year. The researcher chooses a random sample of 650 exams in his state. The average score is 475 (so x x = 475). Estimate the mean SAT math score in this state for this year. We answer this question by computing and interpreting a confidence interval.
Checking conditions: From our work in "Distribution of Sample Means," we know that a normal model is a good fit for the distribution of sample means from random samples if one of two conditions is met: The population of individual values is normal (in which case the sample size is not important). If we do not know if the population of individual values is normal, then we must have a large sample size (more than 30). Because we assume that the distribution of individual SAT math scores is normal in this example, a normal model is also a good fit for the distribution of sample means. Even if the population distribution had not been normal, the sample size is large enough that the normal distribution would still apply to the sample means. So we can use the confidence interval formula given above. Finding the margin of error: Keep in mind that the sample mean, x x, is only a single-value estimate for the population mean, μ. Because it comes from a random sample, we expect there to be some error in the estimate. But how much error should we expect? We know that the sample distribution of means is approximately normal because conditions are met. Recall that in a normal model, 95% of the values fall within 2 standard deviations of the mean, so we use 2 standard errors for our margin of error. This was part of the empirical rule from the module Probability and Probability Distribution.
Conclusion: The following material comes from Concepts in Statistics written by the Open Learning Initiative (OLI) We are 95% confident that the mean SAT math score in this state this year is between 467.2 and 484.8. Recall from our previous work that being 95% confident means this method, in the long run, captures the true population mean (μ) about 95% of the time. Summary If we want to estimate µ, a population mean, we want to calculate a confidence interval. The 95% confidence interval is: We say we are 95% confident that this interval contains µ, which means that in the long run, 95% of these confidence intervals contain µ.we can use this formula only if a normal model is a good fit for the sampling distribution of sample means. If the sample size is large (n > 30), we can use a normal model. If the sample size is not greater than 30, then we can use a normal model only if the variable is normally distributed in the population. As always, we must have a random sample. If the sample is not random, we cannot use it to estimate µ. What Does 95% Confident Really Mean? The following activity revisits the concept of 95% confident, a probability statement that is often misinterpreted. Comment In our work with confidence intervals for estimating a population mean, µ, we require the population standard deviation, σ, to be known. In practice, σ usually is unknown. However, in some situations, especially when a lot of research has been done on the quantitative variable whose mean we are estimating (such as IQ, height, weight, scores on standardized tests), it is reasonable to assume that σ is known. On the next page, we learn how to proceed when σ is unknown. Content by the Open Learning Initiative and licensed under CC BY.
Module 28 - Estimating a Population Mean (2 of 3) Introduction to the T-Model Here is the formula for the T-score. We also include the z-score for comparison. The formulas are very similar. The distribution of z-scores is the standard normal curve, with mean of 0 and standard deviation of 1. The distribution of T-scores depends on the sample size, n. There is a different T-model for every n. So the T-model is a family of curves. Instead of referring to n to specify which T-model to use, we refer to the degrees of freedom, or df for short. For Topics 10.2 and 10.3, the number of degrees of freedom is 1 less than the sample size. That is, df = n 1.
In summary, a normal model is defined by its mean and standard deviation. A T-model is a family of curves defined by the degrees of freedom. Let s take a look at a few T-model curves (for various df) to see how they compare to the normal model. We can see from the picture that as df grows, the T-model gets closer to the standard normal model. Similarities between T-model and standard normal model: Symmetric with a central peak, bell-shaped. Centered at 0. The larger the degrees of freedom, the closer the T-model is to the standard normal model. Difference between T-model and standard normal model: The T-model has more spread than the standard normal model. The T-model has more probability in the tails and less in the center than the standard normal model. We can see this in the fatter tails and lower central peak of the T-model.
When is a T-model a good fit for the sampling distribution of sample means? Check these conditions before using the T-model: Use the T-model if σ (the population standard deviation) is unknown. If σ is known, then use the normal model instead of the T-model. Use the T-model if variable values are normally distributed in the population. If this is not true, then make sure the sample size is large (more than 30). EXAMPLE Cable Strength A group of engineers developed a new design for a steel cable. They need to estimate the amount of weight the cable can hold. The weight limit will be reported on cable packaging. The engineers take a random sample of 45 cables and apply weights to each of them until they break. The mean breaking weight for the 45 cables is the breaking weight for the sample is s = 15.1 lb. = 768.2 lb. The standard deviation of What should the engineers report as the mean amount of weight held by this type of cable? Let s use these sample statistics to construct a 95% confidence interval for the mean breaking weight of this type of cable. Checking conditions: Since we do not know the standard deviation of breaking weights of all of the cables (the population parameter σ), we use the sample standard deviation (s) as an approximation for σ. Since we don t know σ, we must use the T-distribution to model the sampling distribution of means. Is the T-model a good fit for the sampling distribution? Yes, because the conditions are met: σ is unknown. The sample size is large enough.
Finding the margin of error: To find the margin of error, we need to find the critical T-value that corresponds to a 95% confidence level. This is just like the critical Z-value when we built confidence intervals for proportions, except that it comes from the T-model instead of the standard normal model. We will use technology to find the critical T-value. There are a number of tools for doing this. Some books will also give you the option to use printed tables of values. Here we will use an applet that gives the T-model based on degrees of freedom. We want the T-values that cut off the central 95% of the area under the curve. It will look as follows. Using the applet, we see that the critical T-value for a 95% confidence interval with 44 degrees of freedom is T c = 2.015, which means our margin of error for this confidence interval is Note: For 95% confidence, the empirical rule approximates the critical Z-value as 2. The empirical rule is based on the normal model. Using the T-model for df = 44, the critical T-value (2.015) is very close to 2. This makes sense because for larger df, the T-model is very close to the standard normal model. We will see that the critical T-value differs more from the critical Z- value when the sample sizes are small.
Finding the confidence interval: We have all the pieces to build the confidence interval. In our example, the confidence interval is Conclusion: We are 95% confident that the mean breaking weight for all cables of this type is between 763.7 lb and 772.7 lb. Confidence intervals at the 95% confidence level are common in practice. But 95% is not the only confidence level we use. Particularly in situations that involve safety issues, such as the previous example, people often prefer to estimate population means with 99% confidence intervals. In the following quiz we'll do some exploration with technology to see how changes in the confidence level affect the confidence interval. Content by the Open Learning Initiative and licensed under CC BY.
Module 28 - Estimating a Population Mean (3 of 3) Structure of a Confidence Interval Let s take a closer look at the parts of the confidence interval. Remember that this is a confidence interval for a population mean. We use this formula when the population standard deviation is unknown. Let s remind ourselves how the confidence interval formula relates to the graph of the confidence interval on a number line.
Note: The sample mean (9 in this example) is at the center of the interval. The margin of error (labeled ME and equal to 1.24 in this example) is the distance that the interval extends to the left and right of the sample mean. The interval width is the length of the entire interval on the number line. The interval width is always twice the margin of error. Let s quickly review how the precision of a confidence interval relates to the margin of error: An interval gives a more precise estimate when the interval is narrower. In other words, the margin of error is smaller. An interval gives a less precise estimate when the interval is wider. In other words, the margin of error is larger. We know that a higher confidence level gives a larger margin of error, so confidence level is also related to precision. Increasing the confidence in our estimate makes the confidence interval wider and therefore less precise.
Decreasing the confidence in our estimate makes the confidence interval narrower, and therefore more precise. Confidence interval estimates are useful when they have the right balance of confidence and precision. Typical confidence levels used in practice are 90%, 95%, and 99%. When we need to be really sure about our estimates, such as in life-and-death situations, we choose a 99% confidence level. So if nothing else changes, we settle for less precise estimates when we need a high level of confidence. In our discussion about the structure of confidence intervals, we said choosing a higher level of confidence means that we sacrifice some precision. This is true only if nothing else changes. But there is one way to keep a high level of confidence without sacrificing precision: Increase the sample size. We investigate the impact of sample size on the confidence interval next. EXAMPLE Cable Strength Revisited
Recall the engineers who are trying to determine the breaking weight of a cable. In that example, we had a random sample of 45 cables with a mean breaking weight of 768.2 lb and a standard deviation of 15.1 lb. From that sample we computed a 95% confidence interval for the mean breaking weight of all such cables. Here are the important numbers we found from that calculation on the previous page: Now let s increase the sample size and investigate the impact on the confidence interval. We calculate the confidence interval for a larger sample of 101 cables (n = 101). Sample size affects our calculations in two ways: The sample size (n) appears in our formula for standard error. The critical T-value depends on degrees of freedom, and df = n - 1. Finding the standard error: We approximate the standard error of all sample means as follows: Note: The standard error is smaller when the sample size is larger. We were expecting this because we know there is less variability in sample means when the samples are larger. Finding the critical T-value:Note: The standard error is smaller when the sample size is larger. We were expecting this because we know there is less variability in sample means when the samples are larger. To find the critical T-value, we use the applet. We set the df to 100 and the central probability to 0.95. We see that the critical T-value is 1.984.
Note: Increasing the sample size decreased the critical T-value (the T-value went from 2.015 to 1.984 when we increased the sample size). You might also notice that both of the critical T- values for 95% confidence are larger than the critical Z-value for 95% confidence, which is approximately 1.96. This makes sense because the T-models are wider than the the standard normal curve. Finding the margin of error. Here is the margin of error calculation: Finding the confidence interval. Here is the confidence interval calculation:
Side-by-side comparison: Let s take a look at these two intervals to study the effects of changing the sample size. Increasing the sample size had the following effects on the confidence interval estimate: Decreased standard error Decreased critical T-value Decreased margin of error and hence decreased the interval width Improved interval precision Comment In the real world, increasing the sample size is not always possible. Sometimes collecting a sample is very expensive. If the study has budgetary constraints, which is usually the case, selecting a larger sample may be too expensive. Content by the Open Learning Initiative and licensed under CC BY.
Module 28 - Wrap Up "Estimating a Population Mean" Content by the Open Learning Initiative and licensed under CC BY.